💡Okay, so I have this problem where I need to look at the squares of numbers starting from 1000, like 1000², 1001², 1002², and so on. Then, for each of these squares, I have to discard the last two digits. After doing that, I need to figure out how many of the first terms in this new sequence form an arithmetic progression.Hmm, let me break this down step by step. First, I need to understand what it means to discard the last two digits of a number. If I have a number, say, 123456, and I discard the last two digits, I'm left with 1234. So, mathematically, this is equivalent to performing integer division by 100. That is, 123456 divided by 100 is 1234.56, and taking the integer part gives me 1234.So, for each term in the original sequence, which is (1000 + n)² where n starts from 0, I need to compute this square and then divide by 100, taking the integer part. Let me write that down:The original term is (1000 + n)². Expanding this, it becomes 1000² + 2*1000*n + n², which simplifies to 1,000,000 + 2000n + n². Then, when I divide this by 100, I get (1,000,000 / 100) + (2000n / 100) + (n² / 100), which is 10,000 + 20n + (n²)/100. Since we're taking the integer part, the term becomes 10,000 + 20n + floor(n² / 100).Wait, no, actually, it's not exactly floor(n² / 100). Because when you divide the entire expression by 100, it's 10,000 + 20n + (n²)/100, and then you take the floor of the whole thing. So, it's floor(10,000 + 20n + (n²)/100). Since 10,000 and 20n are integers, the only fractional part comes from (n²)/100. Therefore, the term is 10,000 + 20n + floor(n² / 100).But wait, actually, if I have 10,000 + 20n + (n²)/100, and I take the floor of that, it's equivalent to 10,000 + 20n + floor(n² / 100). Because 10,000 and 20n are integers, so the floor only affects the fractional part from (n²)/100.So, the new sequence is 10,000 + 20n + floor(n² / 100). Now, I need to check when this sequence forms an arithmetic progression. An arithmetic progression has a constant difference between consecutive terms. So, let's compute the difference between consecutive terms.Let me denote the new term as b_n = 10,000 + 20n + floor(n² / 100). Then, the difference between b_{n+1} and b_n is:b_{n+1} - b_n = [10,000 + 20(n+1) + floor((n+1)² / 100)] - [10,000 + 20n + floor(n² / 100)]Simplifying this, the 10,000 cancels out, and we have:20(n+1) - 20n + floor((n+1)² / 100) - floor(n² / 100) = 20 + [floor((n+1)² / 100) - floor(n² / 100)]So, the difference is 20 plus the difference between floor((n+1)² / 100) and floor(n² / 100). For the sequence to be an arithmetic progression, this difference must be constant, which would mean that floor((n+1)² / 100) - floor(n² / 100) must be zero. Otherwise, the difference would change.So, when is floor((n+1)² / 100) - floor(n² / 100) equal to zero? Let's compute this difference:floor((n+1)² / 100) - floor(n² / 100) = floor((n² + 2n + 1)/100) - floor(n² / 100)This difference is equal to zero if (n² + 2n + 1)/100 - n²/100 < 1, which simplifies to (2n + 1)/100 < 1. So, 2n + 1 < 100, which means 2n < 99, so n < 49.5. Since n is an integer, this holds for n ≤ 49.Wait, but that seems too large. Let me think again. Because floor((n+1)² / 100) - floor(n² / 100) can be either 0 or 1, depending on whether (n+1)² crosses a multiple of 100 when moving from n² to (n+1)².So, if (n+1)² is less than the next multiple of 100 after n², then the floor function doesn't increase, so the difference is zero. Otherwise, it increases by 1.So, the difference floor((n+1)² / 100) - floor(n² / 100) is 1 if (n+1)² is a multiple of 100, otherwise 0.Wait, no, that's not exactly right. It's 1 if (n+1)² is greater than or equal to the next multiple of 100 after n². So, more precisely, if (n+1)² - n² ≥ 100, then the floor function increases by at least 1. But (n+1)² - n² = 2n + 1. So, if 2n + 1 ≥ 100, then (n+1)² is at least 100 more than n², so floor((n+1)² / 100) - floor(n² / 100) ≥ 1. But actually, it could be more than 1 if 2n + 1 is more than 100.Wait, no, because (n+1)² = n² + 2n + 1. So, if 2n + 1 is less than 100, then (n+1)² is less than n² + 100, so floor((n+1)² / 100) = floor(n² / 100) + floor((2n + 1)/100). But since 2n + 1 is less than 100, floor((2n + 1)/100) is 0. So, the difference is 0.If 2n + 1 is exactly 100, then floor((n+1)² / 100) = floor(n² / 100) + 1, so the difference is 1.If 2n + 1 is greater than 100, then floor((n+1)² / 100) = floor(n² / 100) + 1 + floor((2n + 1 - 100)/100). But since 2n + 1 is just over 100, say 101, then floor((2n + 1)/100) = 1, so the difference is 1.Wait, maybe I'm overcomplicating. Let's think about when (n+1)² crosses a multiple of 100. So, for each n, if (n+1)² is a multiple of 100, then floor((n+1)² / 100) increases by 1 compared to floor(n² / 100). Otherwise, it stays the same.So, the difference floor((n+1)² / 100) - floor(n² / 100) is 1 if (n+1)² is a multiple of 100, otherwise 0.Therefore, the difference b_{n+1} - b_n is 20 + 1 = 21 when (n+1)² is a multiple of 100, and 20 + 0 = 20 otherwise.So, for the sequence to be an arithmetic progression, the difference must be constant. Therefore, we need that floor((n+1)² / 100) - floor(n² / 100) is always 0. That is, (n+1)² is never a multiple of 100.So, we need to find the values of n for which (n+1)² is not a multiple of 100. That is, n+1 is not a multiple of 10, because 10² = 100. So, n+1 ≠ 10k for any integer k.Therefore, as long as n+1 is not a multiple of 10, the difference remains 20, and the sequence is an arithmetic progression. Once n+1 becomes a multiple of 10, the difference increases by 1, making it 21, thus breaking the arithmetic progression.So, the sequence will form an arithmetic progression until n+1 is less than 10, because when n+1 = 10, which is a multiple of 10, the difference becomes 21. Therefore, n can be from 0 up to 8, because when n=9, n+1=10, which is a multiple of 10.Wait, let me check that. If n starts at 0, then the first term is n=0: (1000 + 0)² = 1000² = 1,000,000. Dividing by 100 gives 10,000. Then n=1: (1001)² = 1,002,001. Dividing by 100 gives 10,020.01, which floors to 10,020. So, the difference is 10,020 - 10,000 = 20.Similarly, n=2: (1002)² = 1,004,004. Divided by 100 is 10,040.04, which floors to 10,040. Difference is 20.Continuing this, for n=3: 10,060.09 → 10,060, difference 20.n=4: 10,080.16 → 10,080, difference 20.n=5: 10,100.25 → 10,100, difference 20.n=6: 10,120.36 → 10,120, difference 20.n=7: 10,140.49 → 10,140, difference 20.n=8: 10,160.64 → 10,160, difference 20.n=9: (1009)² = 1,018,081. Divided by 100 is 10,180.81, which floors to 10,180. Difference from previous term (10,160) is 20.Wait, so n=9 still gives a difference of 20. Then n=10: (1010)² = 1,020,100. Divided by 100 is 10,201.00, which is 10,201. The previous term was 10,180, so the difference is 21.Ah, so at n=10, the difference becomes 21, which breaks the arithmetic progression.Therefore, the sequence forms an arithmetic progression from n=0 up to n=9, which is 10 terms. Then, at n=10, the difference increases, so the progression is broken.Wait, but when n=9, the term is 10,180, and n=10 is 10,201. So, the difference is 21, which is different from the previous difference of 20. Therefore, the arithmetic progression holds for the first 10 terms, from n=0 to n=9.So, the answer should be 10 terms.But let me double-check. Let's list out the terms:n=0: 1000² = 1,000,000 → 10,000n=1: 1001² = 1,002,001 → 10,020n=2: 1002² = 1,004,004 → 10,040n=3: 1003² = 1,006,009 → 10,060n=4: 1004² = 1,008,016 → 10,080n=5: 1005² = 1,010,025 → 10,100n=6: 1006² = 1,012,036 → 10,120n=7: 1007² = 1,014,049 → 10,140n=8: 1008² = 1,016,064 → 10,160n=9: 1009² = 1,018,081 → 10,180n=10: 1010² = 1,020,100 → 10,201So, from n=0 to n=9, the terms are 10,000; 10,020; 10,040; 10,060; 10,080; 10,100; 10,120; 10,140; 10,160; 10,180. Each time, the difference is 20, so it's an arithmetic progression.At n=10, the term is 10,201, which is 21 more than the previous term, so the progression breaks.Therefore, the number of terms in the resulting sequence that form an arithmetic progression is 10.

💡Okay, so I have this problem where I need to find the number of distinct values of ( z = |x - y| ) where ( x ) and ( y ) are four-digit integers, and ( y ) is the reverse of ( x ). Both ( x ) and ( y ) are between 1000 and 9999, inclusive. First, I need to understand what it means for ( y ) to be the reverse of ( x ). If ( x ) is a four-digit number, say ( x = 1000a + 100b + 10c + d ), then ( y ) would be ( 1000d + 100c + 10b + a ). So, ( a ) is the thousands digit of ( x ), ( b ) is the hundreds digit, ( c ) is the tens digit, and ( d ) is the units digit. When reversed, ( d ) becomes the thousands digit of ( y ), ( c ) becomes the hundreds digit, and so on.Now, I need to compute ( z = |x - y| ). Let me write out the expression for ( z ):[z = |(1000a + 100b + 10c + d) - (1000d + 100c + 10b + a)|]Simplifying the expression inside the absolute value:[z = |1000a + 100b + 10c + d - 1000d - 100c - 10b - a|]Combine like terms:[z = |(1000a - a) + (100b - 10b) + (10c - 100c) + (d - 1000d)|][z = |999a + 90b - 90c - 999d|]Factor out common terms:[z = |999(a - d) + 90(b - c)|]I can factor out 99 from both terms:[z = 99|10(a - d) + (b - c)|]So, ( z ) is 99 times the absolute value of ( 10(a - d) + (b - c) ). Now, I need to find how many distinct values ( z ) can take. Since ( z ) is 99 times some integer, the number of distinct ( z ) values depends on how many distinct values ( |10(a - d) + (b - c)| ) can take.Let me analyze ( 10(a - d) + (b - c) ). - ( a ) and ( d ) are digits from 1 to 9 (since they are the first digits of four-digit numbers).- ( b ) and ( c ) are digits from 0 to 9.So, ( a - d ) can range from ( 1 - 9 = -8 ) to ( 9 - 1 = 8 ).Similarly, ( b - c ) can range from ( 0 - 9 = -9 ) to ( 9 - 0 = 9 ).Therefore, ( 10(a - d) ) can range from ( 10 times (-8) = -80 ) to ( 10 times 8 = 80 ).And ( b - c ) can range from -9 to 9.Adding these together, ( 10(a - d) + (b - c) ) can range from ( -80 - 9 = -89 ) to ( 80 + 9 = 89 ).So, the expression inside the absolute value can take integer values from -89 to 89. However, since we take the absolute value, ( |10(a - d) + (b - c)| ) can take values from 0 to 89.Now, I need to check if every integer value between 0 and 89 is achievable. Let's consider the possible values:- The minimum value is 0, which occurs when ( 10(a - d) + (b - c) = 0 ).- The maximum value is 89, which occurs when ( a = 9, d = 1, b = 9, c = 0 ).- Similarly, the minimum negative value is -89, but since we take the absolute value, it becomes 89.Now, I need to ensure that every integer between 0 and 89 can be achieved. Let me think about how ( 10(a - d) + (b - c) ) can produce each integer in this range. For any integer ( k ) between 0 and 89, can I find digits ( a, b, c, d ) such that ( 10(a - d) + (b - c) = k )?Yes, because ( a - d ) can be adjusted to contribute the tens place, and ( b - c ) can be adjusted to contribute the units place. For example, to get ( k = 1 ):- Let ( a - d = 0 ), then ( b - c = 1 ). So, choose ( a = d ), and ( b = c + 1 ).To get ( k = 10 ):- Let ( a - d = 1 ), then ( b - c = 0 ). So, choose ( a = d + 1 ), and ( b = c ).Similarly, for ( k = 15 ):- Let ( a - d = 1 ), then ( b - c = 5 ). So, choose ( a = d + 1 ), and ( b = c + 5 ).This pattern can be extended to all values from 0 to 89. Therefore, ( |10(a - d) + (b - c)| ) can take any integer value from 0 to 89, inclusive. Since ( z = 99 times text{(this value)} ), the distinct values of ( z ) are ( 99 times 0, 99 times 1, 99 times 2, ldots, 99 times 89 ).However, ( z ) is defined as the absolute difference, so ( z ) must be non-negative. But wait, when ( |10(a - d) + (b - c)| = 0 ), ( z = 0 ). However, ( x ) and ( y ) are both four-digit numbers, and if ( x = y ), then ( z = 0 ). But the problem states that ( x ) and ( y ) are between 1000 and 9999, inclusive, but it doesn't specify that ( x ) and ( y ) must be different. So, ( z = 0 ) is possible when ( x ) is a palindrome (reads the same forwards and backwards).But wait, the problem says "both integers between 1000 and 9999, inclusive", so ( x ) and ( y ) can be the same if ( x ) is a palindrome. Therefore, ( z = 0 ) is a valid value.However, I need to check if all values from 0 to 89 are achievable. Wait, let me think again. When ( a - d ) is 0, ( b - c ) can be from -9 to 9, so ( 10(a - d) + (b - c) ) can be from -9 to 9, but when ( a - d ) is 1, it can be from 10 - 9 = 1 to 10 + 9 = 19, and so on.But does this cover all integers from 0 to 89? Let's see:- For ( a - d = 0 ), ( b - c ) can be from -9 to 9, so the expression can be from -9 to 9.- For ( a - d = 1 ), the expression can be from 10 - 9 = 1 to 10 + 9 = 19.- For ( a - d = 2 ), it can be from 20 - 9 = 11 to 20 + 9 = 29.- ...- For ( a - d = 8 ), it can be from 80 - 9 = 71 to 80 + 9 = 89.Wait, so when ( a - d ) increases by 1, the range shifts by 10, but overlaps with the previous range. For example, when ( a - d = 0 ), we cover -9 to 9, but when ( a - d = 1 ), we cover 1 to 19. So, 1 to 9 is covered by both ( a - d = 0 ) and ( a - d = 1 ). Similarly, when ( a - d = 1 ), we cover up to 19, and ( a - d = 2 ) covers 11 to 29, overlapping with 11-19.This overlapping means that every integer from 0 to 89 is covered. Wait, but when ( a - d = 0 ), the expression can be from -9 to 9, but taking absolute value, it becomes 0 to 9. Then, when ( a - d = 1 ), the expression can be from 1 to 19, so absolute values from 1 to 19. Similarly, ( a - d = 2 ) gives absolute values from 11 to 29, and so on.Wait, but when ( a - d = 0 ), the expression can be negative, but taking absolute value, it becomes 0 to 9. Then, for positive values, ( a - d ) can be from 1 to 8, giving expressions from 1 to 89.So, combining all, the absolute values can be from 0 to 89, but with some gaps? Or is it continuous?Wait, no, because when ( a - d = 1 ), the expression can be from 1 to 19, so absolute values 1 to 19. When ( a - d = 2 ), it's 11 to 29, so absolute values 11 to 29. So, the ranges overlap, but do they cover all numbers?Wait, let's see:- ( a - d = 0 ): 0 to 9- ( a - d = 1 ): 1 to 19- ( a - d = 2 ): 11 to 29- ( a - d = 3 ): 21 to 39- ...- ( a - d = 8 ): 71 to 89So, each subsequent ( a - d ) increases the lower bound by 10 and the upper bound by 10. The overlap between each range ensures that every number from 0 to 89 is covered.For example:- 0 is covered by ( a - d = 0 )- 1 is covered by ( a - d = 0 ) and ( a - d = 1 )- 10 is covered by ( a - d = 1 ) (since ( 10(a - d) = 10 ), ( b - c = 0 ))- 11 is covered by ( a - d = 1 ) and ( a - d = 2 )- 20 is covered by ( a - d = 2 ) (since ( 10(a - d) = 20 ), ( b - c = 0 ))- And so on, up to 89.Therefore, every integer from 0 to 89 is achievable, meaning ( |10(a - d) + (b - c)| ) can take 90 distinct values (including 0).Thus, ( z = 99 times text{value} ) will also have 90 distinct values, since multiplying by 99 doesn't change the number of distinct values, just scales them.Wait, but is 0 considered a distinct value? The problem says ( z = |x - y| ), and ( x ) and ( y ) can be the same, so ( z = 0 ) is possible. Therefore, 0 is included.So, the total number of distinct ( z ) values is 90.But wait, let me double-check. If ( a - d ) can be from -8 to 8, and ( b - c ) from -9 to 9, then ( 10(a - d) + (b - c) ) can be from -89 to 89, but taking absolute value, it's 0 to 89. So, 90 distinct values.Yes, that seems correct.Therefore, the number of distinct values of ( z ) is 90.

💡Okay, so I have this problem about Jack's test score. The class mean was 60, and the standard deviation was 10. Jack's score is within the top 10% of his class, and it's also within 2 standard deviations of the mean. I need to find the lowest score he could have received.First, I need to understand what it means to be within the top 10%. That probably means his score is higher than 90% of the class. So, I think I need to find the score that separates the top 10% from the rest. I remember something about z-scores and percentiles. Maybe I can use a z-table or something like that.The z-score formula is z = (X - mean) / standard deviation. I think if I can find the z-score that corresponds to the 90th percentile, I can then find the actual score. I recall that the z-score for the 90th percentile is around 1.28. Let me double-check that. Yeah, I think it's approximately 1.28.So, if I plug that into the formula: 1.28 = (X - 60) / 10. Solving for X, I get X = 1.28 * 10 + 60, which is 12.8 + 60 = 72.8. So, the score that marks the top 10% is about 72.8.Now, the other condition is that Jack's score is within 2 standard deviations of the mean. The mean is 60, and the standard deviation is 10, so 2 standard deviations above the mean is 60 + 20 = 80. So, Jack's score has to be between 40 and 80, but since we're talking about the top 10%, it's actually between 72.8 and 80.But wait, the question is asking for the lowest score he could have received. So, the lowest score within the top 10% and within 2 standard deviations would be the higher of the two lower bounds, which is 72.8. Since test scores are usually whole numbers, I guess we round up to 73.Let me make sure I didn't make any mistakes. I calculated the 90th percentile score as 72.8, and the upper bound of 2 standard deviations is 80. So, the lowest score Jack could have is 73. Yeah, that seems right.

💡Okay, so I have this problem where I need to find the interval where the function ( f(x) = frac{1}{3}x^3 - 4x + 4 ) is decreasing. Hmm, I remember from my calculus class that to determine where a function is increasing or decreasing, I need to look at its first derivative. If the derivative is positive, the function is increasing, and if it's negative, the function is decreasing. Alright, let me start by finding the derivative of ( f(x) ). The function is a polynomial, so I can differentiate term by term. The derivative of ( frac{1}{3}x^3 ) is ( x^2 ) because I multiply the exponent by the coefficient and then reduce the exponent by one. Then, the derivative of ( -4x ) is just ( -4 ), since the exponent of x is 1, so 1 times -4 is -4. The derivative of the constant term, 4, is zero because the slope of a constant function is flat. So putting that all together, the derivative ( f'(x) ) should be ( x^2 - 4 ). Let me write that down: ( f'(x) = x^2 - 4 ). Okay, now I need to figure out where this derivative is negative because that's where the function is decreasing. So I need to solve the inequality ( x^2 - 4 < 0 ). Hmm, solving ( x^2 - 4 < 0 ). I can factor this quadratic expression. It looks like a difference of squares because ( x^2 ) is a square of x and 4 is a square of 2. So, ( x^2 - 4 ) factors into ( (x - 2)(x + 2) ). So the inequality becomes ( (x - 2)(x + 2) < 0 ). Now, I need to find the values of x where this product is less than zero, meaning it's negative. To do that, I can use a sign chart or test intervals around the critical points. The critical points here are where each factor equals zero, which is at x = 2 and x = -2. These points divide the real number line into three intervals: 1. ( (-infty, -2) )2. ( (-2, 2) )3. ( (2, infty) )I need to test each interval to see whether the product ( (x - 2)(x + 2) ) is positive or negative there. Starting with the first interval ( (-infty, -2) ). Let me pick a test point, say x = -3. Plugging into ( (x - 2)(x + 2) ), we get ( (-3 - 2)(-3 + 2) = (-5)(-1) = 5 ), which is positive. So the product is positive in this interval.Next, the interval ( (-2, 2) ). Let's choose x = 0 as the test point. Plugging in, we get ( (0 - 2)(0 + 2) = (-2)(2) = -4 ), which is negative. So the product is negative in this interval.Lastly, the interval ( (2, infty) ). Let's take x = 3. Plugging in, we get ( (3 - 2)(3 + 2) = (1)(5) = 5 ), which is positive. So the product is positive in this interval.So, from this analysis, the inequality ( (x - 2)(x + 2) < 0 ) holds true only in the interval ( (-2, 2) ). That means the derivative ( f'(x) ) is negative in this interval, so the original function ( f(x) ) is decreasing here.Wait, but let me double-check. Sometimes, when dealing with inequalities, especially with quadratics, it's easy to make a mistake with the signs. So let me verify by plugging in another point in each interval just to be sure.For ( (-infty, -2) ), I'll pick x = -4. Then, ( (-4 - 2)(-4 + 2) = (-6)(-2) = 12 ), which is positive. Okay, that's consistent with my earlier result.For ( (-2, 2) ), let me try x = 1. ( (1 - 2)(1 + 2) = (-1)(3) = -3 ), which is negative. Still consistent.For ( (2, infty) ), let me take x = 5. ( (5 - 2)(5 + 2) = (3)(7) = 21 ), which is positive. Yep, that's the same as before.So, I'm confident that the derivative is negative only between -2 and 2. Therefore, the function ( f(x) ) is decreasing on the interval ( (-2, 2) ).Wait a second, let me think about the endpoints. At x = -2 and x = 2, the derivative is zero. So, at those points, the function has horizontal tangents. But since the question is about where the function is decreasing, which is where the derivative is negative, those endpoints are not included. So, the interval is open, not closed.Just to make sure, let me visualize the graph of the function. It's a cubic function, so it should have one local maximum and one local minimum. The derivative is a quadratic, which is a parabola opening upwards because the coefficient of ( x^2 ) is positive. The parabola crosses the x-axis at x = -2 and x = 2. So, between these two roots, the parabola is below the x-axis, meaning the derivative is negative, hence the function is decreasing. Outside of these roots, the parabola is above the x-axis, so the derivative is positive, meaning the function is increasing.Therefore, the function decreases from x = -2 to x = 2 and increases otherwise. So, the interval where it's decreasing is indeed ( (-2, 2) ).I think that's thorough enough. I've checked my work multiple times, tested points in each interval, and even thought about the graph to confirm. I don't see any mistakes in my reasoning.**Final Answer**The interval where the function is decreasing is boxed{A}.

💡Okay, so I'm trying to solve this problem about an ellipse. It has two parts. Let me start with part (1).First, the ellipse is given by the equation ( frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ) where ( a > b > 0 ). The eccentricity is given as ( frac{sqrt{6}}{3} ). I remember that the eccentricity ( e ) of an ellipse is related to ( a ) and ( c ) (the distance from the center to each focus) by the formula ( e = frac{c}{a} ). So, ( c = a cdot e ).Given ( e = frac{sqrt{6}}{3} ), so ( c = a cdot frac{sqrt{6}}{3} ). Also, I recall that for an ellipse, ( c^2 = a^2 - b^2 ). So, if I can find ( a ) and ( b ), I can write the equation of the ellipse.Now, the problem mentions a line ( l ) that intersects the x-axis at point ( E ) and intersects the ellipse at points ( A ) and ( B ). When line ( l ) is perpendicular to the x-axis (so it's a vertical line), and point ( E ) is the right focus of the ellipse, the length of chord ( AB ) is ( frac{2sqrt{6}}{3} ).Since the line is vertical and passes through the right focus, the x-coordinate of point ( E ) is ( c ). So, the equation of line ( l ) is ( x = c ). This line intersects the ellipse at points ( A ) and ( B ). To find the coordinates of ( A ) and ( B ), I can substitute ( x = c ) into the ellipse equation.Substituting ( x = c ) into ( frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ), we get:( frac{c^2}{a^2} + frac{y^2}{b^2} = 1 )We know ( c^2 = a^2 - b^2 ), so substituting that in:( frac{a^2 - b^2}{a^2} + frac{y^2}{b^2} = 1 )Simplifying:( 1 - frac{b^2}{a^2} + frac{y^2}{b^2} = 1 )Subtracting 1 from both sides:( -frac{b^2}{a^2} + frac{y^2}{b^2} = 0 )Rearranging:( frac{y^2}{b^2} = frac{b^2}{a^2} )Multiplying both sides by ( b^2 ):( y^2 = frac{b^4}{a^2} )Taking square roots:( y = pm frac{b^2}{a} )So, the points ( A ) and ( B ) have coordinates ( (c, frac{b^2}{a}) ) and ( (c, -frac{b^2}{a}) ). The length of chord ( AB ) is the distance between these two points, which is ( 2 cdot frac{b^2}{a} ). According to the problem, this length is ( frac{2sqrt{6}}{3} ). So:( 2 cdot frac{b^2}{a} = frac{2sqrt{6}}{3} )Dividing both sides by 2:( frac{b^2}{a} = frac{sqrt{6}}{3} )So, ( b^2 = frac{sqrt{6}}{3} cdot a ). Let me write that down.Earlier, we had ( c = a cdot frac{sqrt{6}}{3} ) and ( c^2 = a^2 - b^2 ). Let's substitute ( c ) into ( c^2 ):( c^2 = left( a cdot frac{sqrt{6}}{3} right)^2 = a^2 cdot frac{6}{9} = frac{2a^2}{3} )So, ( c^2 = frac{2a^2}{3} ). But ( c^2 = a^2 - b^2 ), so:( frac{2a^2}{3} = a^2 - b^2 )Subtracting ( frac{2a^2}{3} ) from both sides:( 0 = a^2 - b^2 - frac{2a^2}{3} )Simplifying:( 0 = frac{a^2}{3} - b^2 )So, ( b^2 = frac{a^2}{3} )But earlier, we had ( b^2 = frac{sqrt{6}}{3} cdot a ). So, setting these equal:( frac{a^2}{3} = frac{sqrt{6}}{3} cdot a )Multiplying both sides by 3:( a^2 = sqrt{6} cdot a )Dividing both sides by ( a ) (assuming ( a neq 0 )):( a = sqrt{6} )So, ( a = sqrt{6} ). Then, ( b^2 = frac{a^2}{3} = frac{6}{3} = 2 ). Therefore, ( b = sqrt{2} ).So, the equation of the ellipse is:( frac{x^2}{6} + frac{y^2}{2} = 1 )Alright, that seems to make sense. Let me just verify.Given ( a = sqrt{6} ), ( b = sqrt{2} ), so ( c = sqrt{a^2 - b^2} = sqrt{6 - 2} = sqrt{4} = 2 ). Wait, but earlier, ( c = a cdot frac{sqrt{6}}{3} = sqrt{6} cdot frac{sqrt{6}}{3} = frac{6}{3} = 2 ). That checks out.Then, when ( x = c = 2 ), substituting into the ellipse equation:( frac{4}{6} + frac{y^2}{2} = 1 ) ⇒ ( frac{2}{3} + frac{y^2}{2} = 1 ) ⇒ ( frac{y^2}{2} = frac{1}{3} ) ⇒ ( y^2 = frac{2}{3} ) ⇒ ( y = pm frac{sqrt{6}}{3} ). So, the length of chord ( AB ) is ( 2 cdot frac{sqrt{6}}{3} = frac{2sqrt{6}}{3} ), which matches the given information. So, part (1) is correct.Now, moving on to part (2). The question is whether there exists a point ( E ) such that ( frac{1}{EA^2} + frac{1}{EB^2} ) is a constant value. If such a point exists, find its coordinates and the constant value.Hmm, this seems more complex. Let me think about how to approach this.First, let's denote point ( E ) as ( (h, 0) ) since it's on the x-axis. The line ( l ) passes through ( E ) and intersects the ellipse at points ( A ) and ( B ). We need to find if there exists an ( h ) such that ( frac{1}{EA^2} + frac{1}{EB^2} ) is constant, regardless of the line ( l ).Wait, but the line ( l ) can be any line passing through ( E ), right? So, for any such line, ( frac{1}{EA^2} + frac{1}{EB^2} ) should be the same constant.Alternatively, maybe the problem is considering all possible lines through ( E ), and we need to see if such an ( E ) exists so that the expression is constant for all such lines.I think that's the case. So, regardless of the slope of line ( l ), as long as it passes through ( E ), the sum ( frac{1}{EA^2} + frac{1}{EB^2} ) remains constant.This seems similar to properties of conic sections and focal points, but I'm not sure. Maybe it's related to the director circle or something else.Let me try to parameterize the line ( l ). Let me assume that line ( l ) has a slope ( m ), so its equation is ( y = m(x - h) ). It passes through ( E(h, 0) ).We can substitute this into the ellipse equation ( frac{x^2}{6} + frac{y^2}{2} = 1 ):( frac{x^2}{6} + frac{[m(x - h)]^2}{2} = 1 )Expanding this:( frac{x^2}{6} + frac{m^2(x^2 - 2hx + h^2)}{2} = 1 )Multiply through by 6 to eliminate denominators:( x^2 + 3m^2(x^2 - 2hx + h^2) = 6 )Expanding:( x^2 + 3m^2x^2 - 6m^2hx + 3m^2h^2 - 6 = 0 )Combine like terms:( (1 + 3m^2)x^2 - 6m^2h x + (3m^2h^2 - 6) = 0 )This is a quadratic equation in ( x ). Let me denote this as:( A x^2 + B x + C = 0 )Where:( A = 1 + 3m^2 )( B = -6m^2h )( C = 3m^2h^2 - 6 )Let the roots be ( x_1 ) and ( x_2 ), which correspond to the x-coordinates of points ( A ) and ( B ). Then, from Vieta's formulas:( x_1 + x_2 = -B/A = frac{6m^2h}{1 + 3m^2} )( x_1 x_2 = C/A = frac{3m^2h^2 - 6}{1 + 3m^2} )Now, points ( A ) and ( B ) lie on the line ( y = m(x - h) ), so their coordinates are ( (x_1, m(x_1 - h)) ) and ( (x_2, m(x_2 - h)) ).We need to compute ( EA^2 ) and ( EB^2 ). Since ( E ) is ( (h, 0) ), the distance squared from ( E ) to ( A ) is:( EA^2 = (x_1 - h)^2 + [m(x_1 - h)]^2 = (x_1 - h)^2 (1 + m^2) )Similarly,( EB^2 = (x_2 - h)^2 (1 + m^2) )Therefore,( frac{1}{EA^2} + frac{1}{EB^2} = frac{1}{(x_1 - h)^2 (1 + m^2)} + frac{1}{(x_2 - h)^2 (1 + m^2)} )Factor out ( frac{1}{1 + m^2} ):( frac{1}{1 + m^2} left( frac{1}{(x_1 - h)^2} + frac{1}{(x_2 - h)^2} right) )Let me compute ( frac{1}{(x_1 - h)^2} + frac{1}{(x_2 - h)^2} ).Note that ( x_1 ) and ( x_2 ) are roots of the quadratic equation, so ( x_1 - h ) and ( x_2 - h ) are roots of the equation obtained by substituting ( x = t + h ) into the quadratic.Let me do that substitution:Let ( x = t + h ). Then,( (1 + 3m^2)(t + h)^2 - 6m^2h(t + h) + (3m^2h^2 - 6) = 0 )Expanding:( (1 + 3m^2)(t^2 + 2ht + h^2) - 6m^2h t - 6m^2h^2 + 3m^2h^2 - 6 = 0 )Multiply out:( (1 + 3m^2)t^2 + 2h(1 + 3m^2)t + (1 + 3m^2)h^2 - 6m^2h t - 6m^2h^2 + 3m^2h^2 - 6 = 0 )Combine like terms:- Coefficient of ( t^2 ): ( 1 + 3m^2 )- Coefficient of ( t ): ( 2h(1 + 3m^2) - 6m^2h = 2h + 6m^2h - 6m^2h = 2h )- Constant term: ( (1 + 3m^2)h^2 - 6m^2h^2 + 3m^2h^2 - 6 )Simplify constant term:( (1 + 3m^2 - 6m^2 + 3m^2)h^2 - 6 = (1)h^2 - 6 = h^2 - 6 )So, the equation becomes:( (1 + 3m^2)t^2 + 2h t + (h^2 - 6) = 0 )So, ( t_1 = x_1 - h ) and ( t_2 = x_2 - h ) are roots of this quadratic equation. Therefore, from Vieta's formulas:( t_1 + t_2 = -frac{2h}{1 + 3m^2} )( t_1 t_2 = frac{h^2 - 6}{1 + 3m^2} )We need ( frac{1}{t_1^2} + frac{1}{t_2^2} ). Let me compute this.( frac{1}{t_1^2} + frac{1}{t_2^2} = frac{t_1^2 + t_2^2}{(t_1 t_2)^2} )First, compute ( t_1^2 + t_2^2 ):( t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2 t_1 t_2 )Substitute the known values:( left( -frac{2h}{1 + 3m^2} right)^2 - 2 cdot frac{h^2 - 6}{1 + 3m^2} )Compute each term:First term: ( frac{4h^2}{(1 + 3m^2)^2} )Second term: ( frac{2(h^2 - 6)}{1 + 3m^2} )So,( t_1^2 + t_2^2 = frac{4h^2}{(1 + 3m^2)^2} - frac{2(h^2 - 6)}{1 + 3m^2} )To combine these, let me write them with a common denominator:( frac{4h^2 - 2(h^2 - 6)(1 + 3m^2)}{(1 + 3m^2)^2} )Expand the numerator:( 4h^2 - 2(h^2 - 6)(1 + 3m^2) )First, expand ( (h^2 - 6)(1 + 3m^2) ):( h^2(1 + 3m^2) - 6(1 + 3m^2) = h^2 + 3h^2 m^2 - 6 - 18m^2 )Multiply by 2:( 2h^2 + 6h^2 m^2 - 12 - 36m^2 )So, the numerator becomes:( 4h^2 - (2h^2 + 6h^2 m^2 - 12 - 36m^2) )Simplify:( 4h^2 - 2h^2 - 6h^2 m^2 + 12 + 36m^2 )Which is:( 2h^2 - 6h^2 m^2 + 12 + 36m^2 )Factor terms:Let me factor terms with ( m^2 ):( -6h^2 m^2 + 36m^2 = m^2(-6h^2 + 36) = m^2(36 - 6h^2) = 6m^2(6 - h^2) )And the constant terms:( 2h^2 + 12 )So, numerator is:( 2h^2 + 12 + 6m^2(6 - h^2) )So, putting it all together:( t_1^2 + t_2^2 = frac{2h^2 + 12 + 6m^2(6 - h^2)}{(1 + 3m^2)^2} )Now, ( (t_1 t_2)^2 = left( frac{h^2 - 6}{1 + 3m^2} right)^2 = frac{(h^2 - 6)^2}{(1 + 3m^2)^2} )Therefore,( frac{1}{t_1^2} + frac{1}{t_2^2} = frac{2h^2 + 12 + 6m^2(6 - h^2)}{(1 + 3m^2)^2} div frac{(h^2 - 6)^2}{(1 + 3m^2)^2} = frac{2h^2 + 12 + 6m^2(6 - h^2)}{(h^2 - 6)^2} )Simplify numerator:( 2h^2 + 12 + 6m^2(6 - h^2) = 2h^2 + 12 + 36m^2 - 6h^2 m^2 )So,( frac{1}{t_1^2} + frac{1}{t_2^2} = frac{2h^2 + 12 + 36m^2 - 6h^2 m^2}{(h^2 - 6)^2} )Now, going back to the expression for ( frac{1}{EA^2} + frac{1}{EB^2} ):( frac{1}{1 + m^2} cdot frac{2h^2 + 12 + 36m^2 - 6h^2 m^2}{(h^2 - 6)^2} )Let me factor numerator:( 2h^2 + 12 + 36m^2 - 6h^2 m^2 = 2(h^2 + 6) + 6m^2(6 - h^2) )Wait, that's similar to earlier steps. Let me see:Alternatively, factor terms with ( m^2 ):( -6h^2 m^2 + 36m^2 = m^2(-6h^2 + 36) = 6m^2(6 - h^2) )And the constant terms:( 2h^2 + 12 )So, numerator is ( 2h^2 + 12 + 6m^2(6 - h^2) )So, the expression becomes:( frac{1}{1 + m^2} cdot frac{2h^2 + 12 + 6m^2(6 - h^2)}{(h^2 - 6)^2} )Let me write this as:( frac{2h^2 + 12 + 6m^2(6 - h^2)}{(1 + m^2)(h^2 - 6)^2} )Now, to have ( frac{1}{EA^2} + frac{1}{EB^2} ) be constant, this expression must be independent of ( m ). So, the dependence on ( m ) must cancel out.Let me denote ( k = m^2 ). Then, the expression becomes:( frac{2h^2 + 12 + 6k(6 - h^2)}{(1 + k)(h^2 - 6)^2} )We need this to be constant for all ( k geq 0 ). Let me denote the constant as ( C ). So,( frac{2h^2 + 12 + 6k(6 - h^2)}{(1 + k)(h^2 - 6)^2} = C )Multiply both sides by denominator:( 2h^2 + 12 + 6k(6 - h^2) = C(1 + k)(h^2 - 6)^2 )Let me expand the right-hand side:( C(h^2 - 6)^2 + Ck(h^2 - 6)^2 )So, the equation becomes:( 2h^2 + 12 + 6k(6 - h^2) = C(h^2 - 6)^2 + Ck(h^2 - 6)^2 )Now, let's collect like terms in ( k ):Left side: ( 6(6 - h^2)k + (2h^2 + 12) )Right side: ( C(h^2 - 6)^2 + C(h^2 - 6)^2 k )So, equate coefficients of ( k ) and the constant terms:For ( k ):( 6(6 - h^2) = C(h^2 - 6)^2 )For constants:( 2h^2 + 12 = C(h^2 - 6)^2 )So, we have two equations:1. ( 6(6 - h^2) = C(h^2 - 6)^2 )2. ( 2h^2 + 12 = C(h^2 - 6)^2 )Let me denote ( D = (h^2 - 6)^2 ). Then, equation 1 becomes:( 6(6 - h^2) = C D )Equation 2 becomes:( 2h^2 + 12 = C D )So, both equal to ( C D ). Therefore:( 6(6 - h^2) = 2h^2 + 12 )Let me solve this equation for ( h^2 ):( 36 - 6h^2 = 2h^2 + 12 )Bring all terms to one side:( 36 - 6h^2 - 2h^2 - 12 = 0 )Simplify:( 24 - 8h^2 = 0 )So,( 8h^2 = 24 )( h^2 = 3 )Thus, ( h = pm sqrt{3} ). So, point ( E ) is at ( (sqrt{3}, 0) ) or ( (-sqrt{3}, 0) ).Now, let's find the constant ( C ). From equation 2:( 2h^2 + 12 = C D )We have ( h^2 = 3 ), so:( 2*3 + 12 = C D )( 6 + 12 = C D )( 18 = C D )But ( D = (h^2 - 6)^2 = (3 - 6)^2 = (-3)^2 = 9 )So,( 18 = C * 9 )Thus,( C = 2 )Therefore, the constant value is 2.So, such points ( E ) exist at ( (pm sqrt{3}, 0) ), and the constant is 2.Let me just verify this result. Suppose ( E ) is at ( (sqrt{3}, 0) ). Let me take a specific line, say, the x-axis itself. Then, points ( A ) and ( B ) would be the intersections of the x-axis with the ellipse.The ellipse equation is ( frac{x^2}{6} + frac{y^2}{2} = 1 ). When ( y = 0 ), ( x^2 = 6 ), so ( x = pm sqrt{6} ). So, points ( A ) and ( B ) are ( (sqrt{6}, 0) ) and ( (-sqrt{6}, 0) ).But wait, line ( l ) passes through ( E(sqrt{3}, 0) ). So, if the line is the x-axis, then ( A ) and ( B ) are ( (sqrt{6}, 0) ) and ( (-sqrt{6}, 0) ). But wait, does the line pass through both ( E ) and these points? Yes, because the x-axis passes through ( E ).So, distances ( EA ) and ( EB ):( EA = | sqrt{6} - sqrt{3} | )( EB = | -sqrt{6} - sqrt{3} | = | -(sqrt{6} + sqrt{3}) | = sqrt{6} + sqrt{3} )Wait, but in this case, ( A ) is ( (sqrt{6}, 0) ) and ( B ) is ( (-sqrt{6}, 0) ). So, ( EA = sqrt{6} - sqrt{3} ) and ( EB = sqrt{6} + sqrt{3} ).Compute ( frac{1}{EA^2} + frac{1}{EB^2} ):First, ( EA^2 = (sqrt{6} - sqrt{3})^2 = 6 + 3 - 2sqrt{18} = 9 - 6sqrt{2} )Wait, that can't be right. Wait, ( (sqrt{6} - sqrt{3})^2 = 6 + 3 - 2sqrt{6}sqrt{3} = 9 - 2sqrt{18} = 9 - 6sqrt{2} ). Similarly, ( EB^2 = (sqrt{6} + sqrt{3})^2 = 6 + 3 + 2sqrt{18} = 9 + 6sqrt{2} ).So,( frac{1}{EA^2} + frac{1}{EB^2} = frac{1}{9 - 6sqrt{2}} + frac{1}{9 + 6sqrt{2}} )To compute this, let me rationalize the denominators:( frac{1}{9 - 6sqrt{2}} = frac{9 + 6sqrt{2}}{(9)^2 - (6sqrt{2})^2} = frac{9 + 6sqrt{2}}{81 - 72} = frac{9 + 6sqrt{2}}{9} = 1 + frac{2sqrt{2}}{3} )Similarly,( frac{1}{9 + 6sqrt{2}} = frac{9 - 6sqrt{2}}{81 - 72} = frac{9 - 6sqrt{2}}{9} = 1 - frac{2sqrt{2}}{3} )Adding them together:( (1 + frac{2sqrt{2}}{3}) + (1 - frac{2sqrt{2}}{3}) = 2 )So, indeed, the sum is 2. That's consistent with our earlier result.Let me test another line, say, a vertical line through ( E(sqrt{3}, 0) ). Wait, but in part (1), when the line is vertical through the focus, the chord length was ( frac{2sqrt{6}}{3} ). Let me see.Wait, in part (1), the vertical line was through the focus at ( (2, 0) ), but here, ( E ) is at ( (sqrt{3}, 0) ). So, the vertical line through ( E ) is ( x = sqrt{3} ). Let's find points ( A ) and ( B ).Substitute ( x = sqrt{3} ) into the ellipse equation:( frac{3}{6} + frac{y^2}{2} = 1 ) ⇒ ( frac{1}{2} + frac{y^2}{2} = 1 ) ⇒ ( frac{y^2}{2} = frac{1}{2} ) ⇒ ( y^2 = 1 ) ⇒ ( y = pm 1 )So, points ( A ) and ( B ) are ( (sqrt{3}, 1) ) and ( (sqrt{3}, -1) ).Compute ( EA ) and ( EB ):Since ( E ) is ( (sqrt{3}, 0) ), ( EA = sqrt{ (sqrt{3} - sqrt{3})^2 + (1 - 0)^2 } = sqrt{0 + 1} = 1 ). Similarly, ( EB = 1 ).Thus, ( frac{1}{EA^2} + frac{1}{EB^2} = frac{1}{1} + frac{1}{1} = 2 ). Again, consistent with the constant value.Let me try another line, say, with slope 1. So, equation ( y = x - sqrt{3} ). Substitute into ellipse:( frac{x^2}{6} + frac{(x - sqrt{3})^2}{2} = 1 )Expand:( frac{x^2}{6} + frac{x^2 - 2sqrt{3}x + 3}{2} = 1 )Multiply through by 6:( x^2 + 3(x^2 - 2sqrt{3}x + 3) = 6 )Expand:( x^2 + 3x^2 - 6sqrt{3}x + 9 = 6 )Combine like terms:( 4x^2 - 6sqrt{3}x + 3 = 0 )Divide by 1:( 4x^2 - 6sqrt{3}x + 3 = 0 )Use quadratic formula:( x = frac{6sqrt{3} pm sqrt{(6sqrt{3})^2 - 4*4*3}}{2*4} )Compute discriminant:( (6sqrt{3})^2 - 48 = 108 - 48 = 60 )So,( x = frac{6sqrt{3} pm sqrt{60}}{8} = frac{6sqrt{3} pm 2sqrt{15}}{8} = frac{3sqrt{3} pm sqrt{15}}{4} )So, points ( A ) and ( B ) have x-coordinates ( frac{3sqrt{3} + sqrt{15}}{4} ) and ( frac{3sqrt{3} - sqrt{15}}{4} ). Their y-coordinates are ( y = x - sqrt{3} ).Compute ( EA^2 ) and ( EB^2 ):For point ( A ):( x_A = frac{3sqrt{3} + sqrt{15}}{4} ), ( y_A = frac{3sqrt{3} + sqrt{15}}{4} - sqrt{3} = frac{3sqrt{3} + sqrt{15} - 4sqrt{3}}{4} = frac{-sqrt{3} + sqrt{15}}{4} )So,( EA^2 = left( frac{3sqrt{3} + sqrt{15}}{4} - sqrt{3} right)^2 + left( frac{-sqrt{3} + sqrt{15}}{4} right)^2 )Simplify:( left( frac{3sqrt{3} + sqrt{15} - 4sqrt{3}}{4} right)^2 + left( frac{-sqrt{3} + sqrt{15}}{4} right)^2 )Which is:( left( frac{-sqrt{3} + sqrt{15}}{4} right)^2 + left( frac{-sqrt{3} + sqrt{15}}{4} right)^2 = 2 left( frac{(-sqrt{3} + sqrt{15})^2}{16} right) )Compute ( (-sqrt{3} + sqrt{15})^2 = 3 + 15 - 2sqrt{45} = 18 - 6sqrt{5} )So,( EA^2 = 2 cdot frac{18 - 6sqrt{5}}{16} = frac{36 - 12sqrt{5}}{16} = frac{9 - 3sqrt{5}}{4} )Similarly, for point ( B ):( x_B = frac{3sqrt{3} - sqrt{15}}{4} ), ( y_B = frac{3sqrt{3} - sqrt{15}}{4} - sqrt{3} = frac{3sqrt{3} - sqrt{15} - 4sqrt{3}}{4} = frac{-sqrt{3} - sqrt{15}}{4} )So,( EB^2 = left( frac{3sqrt{3} - sqrt{15}}{4} - sqrt{3} right)^2 + left( frac{-sqrt{3} - sqrt{15}}{4} right)^2 )Simplify:( left( frac{3sqrt{3} - sqrt{15} - 4sqrt{3}}{4} right)^2 + left( frac{-sqrt{3} - sqrt{15}}{4} right)^2 )Which is:( left( frac{-sqrt{3} - sqrt{15}}{4} right)^2 + left( frac{-sqrt{3} - sqrt{15}}{4} right)^2 = 2 left( frac{(-sqrt{3} - sqrt{15})^2}{16} right) )Compute ( (-sqrt{3} - sqrt{15})^2 = 3 + 15 + 2sqrt{45} = 18 + 6sqrt{5} )So,( EB^2 = 2 cdot frac{18 + 6sqrt{5}}{16} = frac{36 + 12sqrt{5}}{16} = frac{9 + 3sqrt{5}}{4} )Now, compute ( frac{1}{EA^2} + frac{1}{EB^2} ):( frac{1}{frac{9 - 3sqrt{5}}{4}} + frac{1}{frac{9 + 3sqrt{5}}{4}} = frac{4}{9 - 3sqrt{5}} + frac{4}{9 + 3sqrt{5}} )Factor out 3 from denominators:( frac{4}{3(3 - sqrt{5})} + frac{4}{3(3 + sqrt{5})} = frac{4}{3} left( frac{1}{3 - sqrt{5}} + frac{1}{3 + sqrt{5}} right) )Combine the fractions:( frac{4}{3} cdot frac{(3 + sqrt{5}) + (3 - sqrt{5})}{(3 - sqrt{5})(3 + sqrt{5})} = frac{4}{3} cdot frac{6}{9 - 5} = frac{4}{3} cdot frac{6}{4} = frac{4}{3} cdot frac{3}{2} = 2 )So, again, the sum is 2. This confirms that regardless of the line through ( E(sqrt{3}, 0) ), the expression ( frac{1}{EA^2} + frac{1}{EB^2} ) is constant and equal to 2.Therefore, the answer to part (2) is that such points ( E ) exist at ( (pm sqrt{3}, 0) ), and the constant value is 2.**Final Answer**(1) The equation of the ellipse is boxed{dfrac{x^2}{6} + dfrac{y^2}{2} = 1}.(2) Such points ( E ) exist at coordinates boxed{(pm sqrt{3}, 0)} and the constant value is boxed{2}.

💡First, I need to determine the total fraction of Linda's savings that was spent on furniture and home appliances. She spent 7/13 on furniture and 3/13 on home appliances, which adds up to 10/13 of her savings.This means that the remaining fraction of her savings, which is 3/13, was spent on the smart TV. The cost of the smart TV is given as 180.Next, I'll set up an equation to find Linda's original savings. Let S represent her original savings. According to the problem:3/13 of S equals 180.So, (3/13) * S = 180.To solve for S, I'll multiply both sides of the equation by 13/3:S = 180 * (13/3).Calculating this gives S = 780.Therefore, Linda's original savings were 780.