💡Okay, so I have this problem here: there's a quadratic polynomial ( x^2 + dx + e ), and it's given that it has exactly one real root. Also, there's a relationship between ( d ) and ( e ): ( d = 2e - 3 ). I need to find the product of all possible values of ( e ).Hmm, let me think. First, I remember that for a quadratic equation ( ax^2 + bx + c = 0 ), the discriminant ( Delta ) is given by ( b^2 - 4ac ). If the discriminant is zero, the quadratic has exactly one real root, which is a repeated root. So, in this case, since the polynomial ( x^2 + dx + e ) has exactly one real root, its discriminant must be zero.Let me write that down. The discriminant ( Delta ) for ( x^2 + dx + e ) is ( d^2 - 4e ). Setting this equal to zero gives:[ d^2 - 4e = 0 ]But I also know that ( d = 2e - 3 ). So, I can substitute ( d ) in the discriminant equation with ( 2e - 3 ). Let me do that:[ (2e - 3)^2 - 4e = 0 ]Now, I need to expand ( (2e - 3)^2 ). Let me compute that:[ (2e - 3)^2 = (2e)^2 - 2 times 2e times 3 + 3^2 = 4e^2 - 12e + 9 ]So, substituting back into the equation:[ 4e^2 - 12e + 9 - 4e = 0 ]Combine like terms. The terms with ( e ) are ( -12e ) and ( -4e ), which add up to ( -16e ). So, the equation becomes:[ 4e^2 - 16e + 9 = 0 ]Now, this is a quadratic equation in terms of ( e ). To find the values of ( e ), I can use the quadratic formula. The quadratic formula states that for an equation ( ax^2 + bx + c = 0 ), the solutions are:[ e = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]In this case, ( a = 4 ), ( b = -16 ), and ( c = 9 ). Plugging these into the formula:[ e = frac{-(-16) pm sqrt{(-16)^2 - 4 times 4 times 9}}{2 times 4} ]Simplify step by step. First, compute the numerator:The first part is ( -(-16) ), which is 16.Next, compute the discriminant inside the square root:[ (-16)^2 = 256 ][ 4 times 4 times 9 = 144 ]So, the discriminant is ( 256 - 144 = 112 ).Therefore, the expression becomes:[ e = frac{16 pm sqrt{112}}{8} ]Simplify ( sqrt{112} ). Let's see, 112 can be factored into 16 and 7, so:[ sqrt{112} = sqrt{16 times 7} = 4sqrt{7} ]So, substituting back:[ e = frac{16 pm 4sqrt{7}}{8} ]I can factor out a 4 from the numerator:[ e = frac{4(4 pm sqrt{7})}{8} ]Simplify the fraction by dividing numerator and denominator by 4:[ e = frac{4 pm sqrt{7}}{2} ]So, the two possible values of ( e ) are ( frac{4 + sqrt{7}}{2} ) and ( frac{4 - sqrt{7}}{2} ).But the problem asks for the product of all possible values of ( e ). Hmm, so I need to multiply these two solutions together.Let me denote the two solutions as ( e_1 ) and ( e_2 ):[ e_1 = frac{4 + sqrt{7}}{2} ][ e_2 = frac{4 - sqrt{7}}{2} ]So, the product ( e_1 times e_2 ) is:[ left( frac{4 + sqrt{7}}{2} right) times left( frac{4 - sqrt{7}}{2} right) ]This looks like the product of a sum and difference, which is ( (a + b)(a - b) = a^2 - b^2 ). Applying that here:[ left( frac{4}{2} right)^2 - left( frac{sqrt{7}}{2} right)^2 ]Wait, actually, let me compute it step by step to avoid confusion.First, multiply the numerators:[ (4 + sqrt{7})(4 - sqrt{7}) = 4^2 - (sqrt{7})^2 = 16 - 7 = 9 ]Then, multiply the denominators:[ 2 times 2 = 4 ]So, the product is:[ frac{9}{4} ]Therefore, the product of all possible values of ( e ) is ( frac{9}{4} ).Alternatively, I remember that for a quadratic equation ( ax^2 + bx + c = 0 ), the product of the roots is ( frac{c}{a} ). In this case, the quadratic equation in ( e ) is ( 4e^2 - 16e + 9 = 0 ). So, the product of the roots is ( frac{9}{4} ), which matches what I found earlier.So, both methods confirm that the product is ( frac{9}{4} ).**Final Answer**The product of all possible values of ( e ) is boxed{dfrac{9}{4}}.

💡Okay, so I have this problem here: If ( a ) and ( b ) are coprime positive integers and ( n ) is an integer, I need to prove that the greatest common divisor of ( a^2 + b^2 - nab ) and ( a + b ) divides ( n + 2 ). Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, let me denote ( d = gcd(a^2 + b^2 - nab, a + b) ). My goal is to show that ( d ) divides ( n + 2 ). So, I need to find some relationship between ( d ) and ( n + 2 ).I remember that the greatest common divisor has a property where ( gcd(x, y) = gcd(x - ky, y) ) for any integer ( k ). Maybe I can use that here. Let me see. If I take ( x = a^2 + b^2 - nab ) and ( y = a + b ), then perhaps I can subtract some multiple of ( y ) from ( x ) to simplify the expression.Let me choose ( k = a - b ) or something like that. Wait, actually, maybe ( k = a + b ) would be better because ( y = a + b ). Let me try that. So, subtracting ( (a + b) times (a + b) ) from ( x ):( x - (a + b)y = (a^2 + b^2 - nab) - (a + b)^2 ).Let me compute that:( (a^2 + b^2 - nab) - (a^2 + 2ab + b^2) = a^2 + b^2 - nab - a^2 - 2ab - b^2 = -nab - 2ab = -ab(n + 2) ).So, now I have ( gcd(-ab(n + 2), a + b) ). Since the gcd is always non-negative, I can ignore the negative sign, so it's ( gcd(ab(n + 2), a + b) ).Now, I know that ( a ) and ( b ) are coprime. That means ( gcd(a, b) = 1 ). I also remember that if two numbers are coprime, then ( gcd(ab, a + b) = 1 ). Let me verify that.Suppose ( d ) divides both ( ab ) and ( a + b ). Then ( d ) divides ( a + b ) and ( ab ). Since ( d ) divides ( a + b ), it also divides any linear combination of ( a ) and ( b ). In particular, ( d ) divides ( a times b - a times b = 0 ), which isn't helpful. Wait, maybe another approach.If ( d ) divides ( ab ) and ( d ) divides ( a + b ), then ( d ) must divide ( a ) and ( b ) individually? Wait, no, not necessarily. Because ( d ) could share factors with both ( a ) and ( b ), but since ( a ) and ( b ) are coprime, ( d ) must be 1. Yeah, that makes sense. So, ( gcd(ab, a + b) = 1 ).Therefore, ( gcd(ab(n + 2), a + b) = gcd(n + 2, a + b) ). So, ( d ) divides ( n + 2 ), since ( d ) is the gcd of ( ab(n + 2) ) and ( a + b ), which simplifies to the gcd of ( n + 2 ) and ( a + b ). Therefore, ( d ) must divide ( n + 2 ).Wait, let me make sure I didn't skip any steps. I subtracted ( (a + b)^2 ) from ( a^2 + b^2 - nab ) and got ( -ab(n + 2) ). Then, I used the property of gcd to rewrite the original gcd as ( gcd(ab(n + 2), a + b) ). Since ( gcd(ab, a + b) = 1 ), this reduces to ( gcd(n + 2, a + b) ). Therefore, ( d ) divides ( n + 2 ).Is there another way to think about this? Maybe using modular arithmetic. If ( d ) divides both ( a^2 + b^2 - nab ) and ( a + b ), then ( a equiv -b mod d ). Let me substitute ( a equiv -b mod d ) into the first expression.So, ( a^2 + b^2 - nab equiv (-b)^2 + b^2 - n(-b)b mod d ).Calculating that:( b^2 + b^2 + nb^2 = (2 + n)b^2 mod d ).Since ( d ) divides ( a^2 + b^2 - nab ), this expression must be congruent to 0 modulo ( d ). So, ( (2 + n)b^2 equiv 0 mod d ).But ( d ) divides ( a + b ), and since ( a ) and ( b ) are coprime, ( d ) must also be coprime with ( b ). Because if ( d ) had a common factor with ( b ), it would also have to divide ( a ), which is coprime with ( b ). Therefore, ( d ) and ( b ) are coprime, so ( d ) must divide ( 2 + n ).That's another way to see it. So, whether I manipulate the expressions directly or use modular arithmetic, I end up with the conclusion that ( d ) divides ( n + 2 ).I think that solidifies the proof. I didn't see any gaps in the logic, and both approaches confirm the result. So, yeah, I feel confident about this now.

💡Alright, let's try to figure out how many of the dwarfs are truthful. So, we have 10 dwarfs, each either always tells the truth or always lies. Each of them likes exactly one type of ice cream: vanilla, chocolate, or fruit. First, Snow White asked those who like vanilla ice cream to raise their hands, and everyone raised their hands. Hmm, that's interesting. So, all 10 dwarfs raised their hands when asked about vanilla. Then, she asked about chocolate, and half of them raised their hands, which would be 5 dwarfs. Finally, she asked about fruit, and only one dwarf raised their hand.Okay, so let's break this down. If all dwarfs raised their hands for vanilla, that means either they all like vanilla, or some of them are lying. But we know each dwarf likes exactly one type of ice cream, so they can't like more than one. So, if a truthful dwarf was asked about vanilla, they would raise their hand only if they actually like vanilla. But a liar would raise their hand even if they don't like vanilla because they always lie.Wait, so if all dwarfs raised their hands for vanilla, that means that the truthful dwarfs who like vanilla raised their hands, and the liars, who don't like vanilla, also raised their hands. So, the number of dwarfs who like vanilla is equal to the number of truthful dwarfs who like vanilla, and the liars, who don't like vanilla, also raised their hands. So, the total number of hands raised for vanilla is the number of truthful dwarfs who like vanilla plus the number of liars.But we don't know how many truthful dwarfs like vanilla. Hmm. Maybe we can set up some equations.Let me denote:- Let T be the number of truthful dwarfs.- Let L be the number of lying dwarfs.- Let V be the number of dwarfs who like vanilla.- Let C be the number of dwarfs who like chocolate.- Let F be the number of dwarfs who like fruit.We know that T + L = 10, since there are 10 dwarfs in total.Also, each dwarf likes exactly one type of ice cream, so V + C + F = 10.From the first question, all 10 dwarfs raised their hands for vanilla. So, the number of truthful dwarfs who like vanilla plus the number of liars who don't like vanilla equals 10.Wait, but the liars would lie about liking vanilla. So, if a liar doesn't like vanilla, they would raise their hand when asked about vanilla. Similarly, if a liar does like vanilla, they would not raise their hand when asked about vanilla. But we don't know how many liars like vanilla.This is getting a bit complicated. Maybe I need to think differently.Let's consider the total number of hands raised across all three questions. For vanilla, 10 hands; for chocolate, 5 hands; for fruit, 1 hand. So, total hands raised = 10 + 5 + 1 = 16.Now, each truthful dwarf will raise their hand exactly once, for their favorite flavor. Each liar, on the other hand, will raise their hand twice because they lie about two flavors they don't like. So, if a liar likes vanilla, they would lie about chocolate and fruit, so they would raise their hand twice. Similarly, if a liar likes chocolate, they would lie about vanilla and fruit, raising their hand twice, and the same for fruit.So, the total number of hands raised can be expressed as T + 2L = 16.We also know that T + L = 10.So, we have two equations:1. T + L = 102. T + 2L = 16Subtracting the first equation from the second, we get:(T + 2L) - (T + L) = 16 - 10Which simplifies to L = 6.So, there are 6 liars. Therefore, the number of truthful dwarfs is T = 10 - L = 10 - 6 = 4.Wait, let me double-check. If there are 4 truthful dwarfs and 6 liars, then the total hands raised would be 4 + 2*6 = 4 + 12 = 16, which matches the total hands raised. That seems consistent.Also, checking the individual questions:- For vanilla, all 10 raised their hands. The truthful dwarfs who like vanilla would raise their hand, and the liars (who don't like vanilla) would also raise their hands. So, the number of truthful dwarfs who like vanilla plus the number of liars equals 10. But we don't know how many truthful dwarfs like vanilla, but since we're only asked for the number of truthful dwarfs, which is 4, that seems okay.Similarly, for chocolate, 5 raised their hands. That would be the truthful dwarfs who like chocolate plus the liars who don't like chocolate. And for fruit, only 1 raised their hand, which would be the truthful dwarf who likes fruit plus the liars who don't like fruit.But since we don't have information about the distribution of preferences among the truthful dwarfs, we can't determine exactly how many like each flavor, but we can determine the number of truthful dwarfs based on the total hands raised.So, I think the answer is 4 truthful dwarfs.

💡Okay, so I have this problem where I need to find the sum of the minimum and maximum possible values of T, given that B² + M² + T² = 2022 and B + M + T = 72. Hmm, let me think about how to approach this.First, I know that when dealing with sums and sums of squares, sometimes it's helpful to use the Cauchy-Schwarz inequality or maybe some kind of optimization technique. But since we have three variables, B, M, and T, maybe I can express B and M in terms of T and then work with just one variable.Let me try that. If B + M + T = 72, then B + M = 72 - T. Similarly, from the sum of squares, B² + M² = 2022 - T². Okay, so now I have B + M and B² + M² in terms of T. Maybe I can use these to find some relationship.I remember that (B + M)² = B² + M² + 2BM. So if I plug in the expressions I have:(72 - T)² = (2022 - T²) + 2BM.Let me compute (72 - T)². That would be 72² - 2*72*T + T², which is 5184 - 144T + T². So:5184 - 144T + T² = 2022 - T² + 2BM.Hmm, let's rearrange this equation to solve for BM. Subtract 2022 - T² from both sides:5184 - 144T + T² - 2022 + T² = 2BM.Simplify the left side:5184 - 2022 is 3162, and T² + T² is 2T², so:2T² - 144T + 3162 = 2BM.Divide both sides by 2:T² - 72T + 1581 = BM.Okay, so BM is equal to T² - 72T + 1581. Now, I need to relate this to something else. Maybe I can use the fact that for real numbers B and M, the discriminant of the quadratic equation they satisfy must be non-negative. That is, if I consider B and M as roots of the quadratic equation x² - (B + M)x + BM = 0, then the discriminant must be greater than or equal to zero.So, the discriminant D is (B + M)² - 4BM. Plugging in the values we have:D = (72 - T)² - 4*(T² - 72T + 1581).Let me compute this:First, expand (72 - T)²: that's 5184 - 144T + T².Then, compute 4*(T² - 72T + 1581): that's 4T² - 288T + 6324.Now, subtract the second expression from the first:D = (5184 - 144T + T²) - (4T² - 288T + 6324).Simplify term by term:5184 - 144T + T² - 4T² + 288T - 6324.Combine like terms:5184 - 6324 = -1140.-144T + 288T = 144T.T² - 4T² = -3T².So, D = -3T² + 144T - 1140.Since the discriminant must be non-negative for real roots B and M, we have:-3T² + 144T - 1140 ≥ 0.Let me multiply both sides by -1 to make it easier, remembering to reverse the inequality:3T² - 144T + 1140 ≤ 0.Divide all terms by 3:T² - 48T + 380 ≤ 0.Now, I need to solve this quadratic inequality. Let me find the roots of the equation T² - 48T + 380 = 0.Using the quadratic formula:T = [48 ± sqrt(48² - 4*1*380)] / 2.Compute the discriminant inside the square root:48² = 2304.4*1*380 = 1520.So, sqrt(2304 - 1520) = sqrt(784) = 28.Therefore, the roots are:T = [48 ± 28] / 2.So, T = (48 + 28)/2 = 76/2 = 38.And T = (48 - 28)/2 = 20/2 = 10.So, the quadratic expression T² - 48T + 380 factors as (T - 10)(T - 38) ≤ 0.To solve this inequality, we can analyze the sign of the expression:- When T < 10, both (T - 10) and (T - 38) are negative, so their product is positive.- When 10 < T < 38, (T - 10) is positive and (T - 38) is negative, so their product is negative.- When T > 38, both factors are positive, so their product is positive.Since the inequality is ≤ 0, the solution is 10 ≤ T ≤ 38.Therefore, the minimum possible value of T is 10, and the maximum possible value is 38. The sum of these two is 10 + 38 = 48.So, the answer should be 48.**Final Answer**boxed{48}

💡Alright, so I need to figure out how many positive three-digit integers, where each digit is at least 5, are divisible by 9. Hmm, okay, let's break this down step by step.First, let's recall that a three-digit number ranges from 100 to 999. But in this case, each digit has to be at least 5. So, the hundreds digit can be 5, 6, 7, 8, or 9. The same goes for the tens and units digits. That means each digit has 5 possible choices.Now, the number has to be divisible by 9. I remember that a number is divisible by 9 if the sum of its digits is a multiple of 9. So, for a three-digit number, the sum of the hundreds, tens, and units digits must be divisible by 9.Let's denote the hundreds digit as H, the tens digit as T, and the units digit as U. So, H, T, U ∈ {5, 6, 7, 8, 9}. We need H + T + U to be a multiple of 9.What's the smallest and largest possible sum of H + T + U? The smallest sum would be 5 + 5 + 5 = 15, and the largest sum would be 9 + 9 + 9 = 27. So, the possible sums that are multiples of 9 within this range are 18 and 27.Wait, 18 and 27? Let me check: 9 × 2 = 18, and 9 × 3 = 27. So, yes, those are the only multiples of 9 between 15 and 27.So, we need to find all three-digit numbers where each digit is at least 5, and the sum of the digits is either 18 or 27.Let's start with the sum being 27. That would mean all three digits are 9, right? Because 9 + 9 + 9 = 27. So, the only number in this case is 999.Now, let's consider the sum being 18. This is a bit more involved because there are multiple combinations of H, T, and U that add up to 18.I think the best way to approach this is to list all possible combinations of H, T, and U where each digit is between 5 and 9, and their sum is 18. Then, for each combination, count how many unique three-digit numbers can be formed.Let's list the combinations:1. 5, 6, 7: 5 + 6 + 7 = 182. 5, 5, 8: 5 + 5 + 8 = 183. 6, 6, 6: 6 + 6 + 6 = 184. 5, 7, 6: This is the same as the first combination, just reordered.5. 5, 8, 5: Same as the second combination.6. 6, 5, 7: Same as the first combination.7. 6, 7, 5: Same as the first combination.8. 7, 5, 6: Same as the first combination.9. 7, 6, 5: Same as the first combination.10. 5, 9, 4: Wait, 4 is less than 5, so this combination is invalid.11. 6, 6, 6: Already listed.12. 7, 7, 4: Again, 4 is invalid.13. 8, 5, 5: Same as the second combination.14. 9, 5, 4: Invalid.15. 9, 6, 3: Invalid.16. 9, 7, 2: Invalid.17. 9, 8, 1: Invalid.18. 9, 9, 0: Invalid.Okay, so after removing duplicates and invalid combinations, we have:1. 5, 6, 72. 5, 5, 83. 6, 6, 6Now, let's count how many unique three-digit numbers each combination can form.For the first combination, 5, 6, 7: All digits are distinct. So, the number of permutations is 3! = 6.For the second combination, 5, 5, 8: Two digits are the same, and one is different. The number of unique permutations is 3 (positions for the digit 8).For the third combination, 6, 6, 6: All digits are the same. So, there's only 1 unique number: 666.Adding these up: 6 + 3 + 1 = 10.But wait, earlier I thought the sum could be 27, which gives only one number: 999. So, in total, we have 10 + 1 = 11 numbers.Hmm, but I thought earlier that the sum could only be 18 or 27. Let me double-check if there are any other sums.Wait, 9 × 1 = 9, but the minimum sum is 15, so 9 is too low. 9 × 2 = 18, which we've covered, and 9 × 3 = 27, which we've also covered. So, no other sums in between.But wait, earlier I thought the answer was 9, but now I'm getting 11. Did I make a mistake?Let me recount the combinations:1. 5, 6, 7: 6 permutations2. 5, 5, 8: 3 permutations3. 6, 6, 6: 1 permutation4. 9, 9, 9: 1 permutationSo, total is 6 + 3 + 1 + 1 = 11.But in my initial thought process, I only considered sum = 18 and got 9 combinations, but now I'm getting 11.Wait, maybe I missed some combinations for sum = 18.Let me try listing all possible combinations again:- 5, 6, 7: sum = 18- 5, 5, 8: sum = 18- 6, 6, 6: sum = 18- 5, 7, 6: same as 5,6,7- 5, 8, 5: same as 5,5,8- 6, 5, 7: same as 5,6,7- 6, 7, 5: same as 5,6,7- 7, 5, 6: same as 5,6,7- 7, 6, 5: same as 5,6,7- 5, 9, 4: invalid- 6, 6, 6: already listed- 7, 7, 4: invalid- 8, 5, 5: same as 5,5,8- 9, 5, 4: invalid- 9, 6, 3: invalid- 9, 7, 2: invalid- 9, 8, 1: invalid- 9, 9, 0: invalidSo, only three unique combinations: 5,6,7; 5,5,8; and 6,6,6.Therefore, the total numbers are 6 + 3 + 1 = 10 for sum = 18, plus 1 for sum = 27, totaling 11.But earlier, I thought the answer was 9. Maybe I made a mistake in the initial thought process.Wait, in the initial thought process, I thought the answer was 9, but now I'm getting 11. Let me check.Wait, in the initial thought process, I considered sum = 18 and sum = 27, but I think I missed the combination 5,5,8 and only counted 5,6,7 and 6,6,6, which would give 6 + 1 = 7, plus 1 for 999, totaling 8. But that's not matching.Wait, I'm getting confused. Let me try a different approach.Instead of listing combinations, maybe I can use stars and bars or some combinatorial method.We need to find the number of solutions to H + T + U = 18 or 27, where H, T, U ≥ 5.But since H, T, U are digits, they can't exceed 9.So, for sum = 18:Let me define H' = H - 5, T' = T - 5, U' = U - 5. Then H', T', U' ≥ 0, and H' + T' + U' = 18 - 15 = 3.Now, the number of non-negative integer solutions to H' + T' + U' = 3 is C(3 + 3 - 1, 3 - 1) = C(5,2) = 10.But wait, this counts all solutions where H', T', U' ≥ 0, but we also have the constraint that H, T, U ≤ 9, which translates to H' ≤ 4, T' ≤ 4, U' ≤ 4.So, we need to subtract the solutions where any of H', T', U' > 4.Using inclusion-exclusion:Total solutions without constraints: C(5,2) = 10.Subtract solutions where H' ≥ 5: Let H'' = H' - 5, then H'' + T' + U' = 3 - 5 = -2, which is impossible. So, no solutions to subtract.Similarly for T' ≥ 5 and U' ≥ 5, no solutions.So, total solutions: 10.But wait, earlier I thought it was 10, but when I listed combinations, I got 10 for sum = 18, plus 1 for sum = 27, totaling 11.But wait, in the stars and bars approach, we're only considering sum = 18, which gives 10 solutions, and sum = 27, which is only 1 solution (999). So, total 11.But earlier, I thought the answer was 9. Maybe I was wrong before.Wait, let me check the combinations again.For sum = 18:- 5,6,7: 6 permutations- 5,5,8: 3 permutations- 6,6,6: 1 permutationTotal: 6 + 3 + 1 = 10.Plus sum = 27: 999.Total: 11.So, the correct answer should be 11.But in the initial thought process, I thought it was 9. Maybe I missed some combinations.Wait, maybe I missed the combination 5,5,8 and 6,6,6, which would add 3 + 1 = 4, plus 6 from 5,6,7, totaling 10, plus 1 for 999, making 11.Yes, so the correct answer is 11.But wait, let me double-check.List all possible three-digit numbers with digits ≥5 and sum of digits divisible by 9.Sum = 18:- 558, 585, 855- 567, 576, 657, 675, 756, 765- 666That's 3 + 6 + 1 = 10 numbers.Sum = 27:- 999Total: 11 numbers.Yes, that's correct.So, the initial thought process had an error, thinking the answer was 9, but actually, it's 11.Therefore, the correct answer is 11.

💡Alright, so I have this problem about a parabola and some points on it. Let me try to parse it step by step.First, the parabola is given by the equation ( y = 4x^2 ). I know that the standard form of a parabola is ( y = ax^2 ), where the vertex is at (0,0) and the focus is at (0, 1/(4a)). So in this case, since a is 4, the focus should be at (0, 1/(4*4)) = (0, 1/16). Let me note that down: vertex ( V_1 ) is (0,0) and focus ( F_1 ) is (0, 1/16).Now, points A and B are on the parabola such that the angle ( angle AV_1B = 90^circ ). I need to find the locus of the midpoint of segment AB, which is called ( mathcal{Q} ). Then, from ( mathcal{Q} ), I have to find its vertex ( V_2 ) and focus ( F_2 ), and finally compute the ratio ( frac{F_1F_2}{V_1V_2} ).Okay, let's break it down.First, let's denote points A and B on the parabola. Since the parabola is ( y = 4x^2 ), any point on it can be written as ( (a, 4a^2) ) and ( (b, 4b^2) ) where a and b are real numbers.So, point A is ( (a, 4a^2) ) and point B is ( (b, 4b^2) ).Given that ( angle AV_1B = 90^circ ). Since ( V_1 ) is the vertex at (0,0), this means that the angle between vectors ( V_1A ) and ( V_1B ) is 90 degrees.I remember that if two vectors are perpendicular, their dot product is zero. So, the vectors ( V_1A ) and ( V_1B ) should have a dot product of zero.Vector ( V_1A ) is from ( V_1 ) to A, which is ( (a - 0, 4a^2 - 0) = (a, 4a^2) ).Similarly, vector ( V_1B ) is ( (b, 4b^2) ).So, their dot product is ( a*b + (4a^2)*(4b^2) ).Wait, no. The dot product is ( (a)(b) + (4a^2)(4b^2) ). Hmm, but that seems a bit complicated. Let me verify.Wait, actually, no. The dot product of two vectors ( (x_1, y_1) ) and ( (x_2, y_2) ) is ( x_1x_2 + y_1y_2 ). So in this case, it's ( a*b + (4a^2)*(4b^2) ).But that would be ( ab + 16a^2b^2 ). Hmm, that seems a bit messy. Maybe I made a mistake.Wait, actually, hold on. The vectors ( V_1A ) and ( V_1B ) are from the vertex to points A and B. So, their coordinates are (a, 4a²) and (b, 4b²). So, the dot product is indeed ( a*b + (4a²)*(4b²) = ab + 16a²b² ).Since they are perpendicular, this should equal zero:( ab + 16a²b² = 0 )Hmm, that seems a bit complicated. Maybe I can factor this:( ab(1 + 16ab) = 0 )So, either ab = 0 or 1 + 16ab = 0.But ab = 0 would mean either a = 0 or b = 0. If a = 0, then point A is the vertex, which is (0,0). Similarly, if b = 0, point B is the vertex. But then, the angle ( angle AV_1B ) would be undefined because both points A and B would coincide with V1. So, ab = 0 is not acceptable because it would make points A or B coincide with V1, which is not meaningful for the angle.Therefore, the other factor must be zero: 1 + 16ab = 0, which gives ab = -1/16.Okay, so ab = -1/16. That's a key relation between a and b.Now, moving on. I need to find the midpoint M of segment AB. The coordinates of M would be the average of the coordinates of A and B.So, the x-coordinate of M is ( frac{a + b}{2} ), and the y-coordinate is ( frac{4a² + 4b²}{2} = 2(a² + b²) ).So, M is ( left( frac{a + b}{2}, 2(a² + b²) right) ).But I need to express this in terms of a single variable to find the locus. Since I have a relation between a and b, which is ab = -1/16, maybe I can express b in terms of a or vice versa.Let me solve for b: b = -1/(16a).So, let's substitute b = -1/(16a) into the coordinates of M.First, the x-coordinate:( frac{a + b}{2} = frac{a - 1/(16a)}{2} = frac{a}{2} - frac{1}{32a} ).Hmm, that seems a bit messy. Maybe instead of substituting, I can express a² + b² in terms of (a + b)².I remember that ( a² + b² = (a + b)² - 2ab ).Since ab = -1/16, then:( a² + b² = (a + b)² - 2*(-1/16) = (a + b)² + 1/8 ).So, substituting back into the y-coordinate:( 2(a² + b²) = 2[(a + b)² + 1/8] = 2(a + b)² + 1/4 ).So, the y-coordinate is ( 2(a + b)² + 1/4 ).Now, let me denote ( h = frac{a + b}{2} ). Then, ( a + b = 2h ).Substituting into the y-coordinate:( 2(2h)² + 1/4 = 2*(4h²) + 1/4 = 8h² + 1/4 ).So, the y-coordinate is ( 8h² + 1/4 ).But h is the x-coordinate of the midpoint M. So, if I let x = h, then y = 8x² + 1/4.Therefore, the locus ( mathcal{Q} ) is the parabola ( y = 8x² + 1/4 ).Okay, so now I have the equation of ( mathcal{Q} ). Let me find its vertex and focus.The standard form of a parabola is ( y = ax² + c ). The vertex is at (0, c). So, in this case, the vertex ( V_2 ) is at (0, 1/4).Now, to find the focus. For a parabola ( y = ax² + c ), the focus is at (0, c + 1/(4a)).Here, a = 8, so 1/(4a) = 1/32.Therefore, the focus ( F_2 ) is at (0, 1/4 + 1/32).Let me compute that:1/4 is 8/32, so 8/32 + 1/32 = 9/32.So, ( F_2 ) is at (0, 9/32).Earlier, I found that ( F_1 ) is at (0, 1/16). Let me write that as 2/32 for easier subtraction.So, ( F_1 ) is at (0, 2/32), ( F_2 ) is at (0, 9/32). The distance between them is 9/32 - 2/32 = 7/32.Wait, hold on. Wait, 9/32 - 2/32 is 7/32? Wait, 9 - 2 is 7, so yes, 7/32.But wait, actually, the distance between two points on the y-axis is just the absolute difference of their y-coordinates. So, yes, |9/32 - 2/32| = 7/32.But wait, hold on. Wait, 1/16 is 2/32, right? So, 9/32 - 2/32 is indeed 7/32.But wait, the problem asks for the ratio ( frac{F_1F_2}{V_1V_2} ).So, first, let me compute ( F_1F_2 ) which is 7/32.Now, ( V_1V_2 ) is the distance between ( V_1 ) at (0,0) and ( V_2 ) at (0,1/4).So, the distance is |1/4 - 0| = 1/4.Convert 1/4 to 32nds: 1/4 = 8/32.So, ( V_1V_2 = 8/32 ).Therefore, the ratio ( frac{F_1F_2}{V_1V_2} = frac{7/32}{8/32} = 7/8 ).Wait, but hold on. Wait, earlier I thought ( F_2 ) was at 9/32, and ( F_1 ) at 2/32, so the distance is 7/32. But the problem says to compute ( frac{F_1F_2}{V_1V_2} ).So, 7/32 divided by 8/32 is 7/8.But wait, in the initial problem, the user had a different calculation, getting 9/8. Hmm, maybe I made a mistake.Wait, let me double-check.First, the focus of ( mathcal{Q} ): ( y = 8x² + 1/4 ). The standard form is ( y = ax² + c ), so the focus is at (0, c + 1/(4a)).Here, a = 8, so 1/(4a) = 1/32. So, c is 1/4, which is 8/32. So, 8/32 + 1/32 = 9/32. So, ( F_2 ) is at (0,9/32). That's correct.( F_1 ) is at (0,1/16) which is 2/32. So, the distance between F1 and F2 is 9/32 - 2/32 = 7/32.( V_1 ) is at (0,0), ( V_2 ) is at (0,1/4) which is 8/32. So, the distance between V1 and V2 is 8/32.Therefore, the ratio is (7/32)/(8/32) = 7/8.Wait, but in the initial problem, the user had a different answer, 9/8. So, maybe I made a mistake somewhere.Wait, let me go back.Wait, in the initial problem, the user wrote:"The focus ( F_2 ) (using ( a = 8 ) in the formula for the focus of ( y = ax^2 ) at ( (0, frac{1}{4a}) )) is at ( (0, frac{1}{4} + frac{1}{32}) )."Wait, hold on. Wait, the formula for the focus of ( y = ax² + c ) is ( (0, c + 1/(4a)) ). So, in this case, c is 1/4, and 1/(4a) is 1/32. So, yes, the focus is at 1/4 + 1/32 = 9/32.So, that's correct.But in the initial problem, the user wrote:"The focus ( F_1 ) of ( mathcal{P} ) is at ( (0, frac{1}{16}) ), using formula ( F = (0, frac{1}{4a}) ) for ( y = 4x^2 )."Yes, that's correct because 1/(4*4) = 1/16.Then, the user calculated:( frac{F_1F_2}{V_1V_2} = frac{frac{9}{32}}{frac{1}{4}} = frac{9}{32} cdot 4 = boxed{frac{9}{8}} ).Wait, hold on. Wait, the user is calculating ( F_1F_2 ) as 9/32, but actually, ( F_1F_2 ) is the distance between F1 and F2, which is 9/32 - 2/32 = 7/32. So, the user made a mistake here.Wait, in the user's solution, they said:"The focus ( F_2 ) is at ( (0, frac{1}{4} + frac{1}{32}) ). The focus ( F_1 ) of ( mathcal{P} ) is at ( (0, frac{1}{16}) )."Then, they calculated ( frac{F_1F_2}{V_1V_2} = frac{frac{9}{32}}{frac{1}{4}} = frac{9}{32} cdot 4 = boxed{frac{9}{8}} ).But wait, ( F_1F_2 ) is the distance between F1 and F2, which is 9/32 - 2/32 = 7/32. So, the user incorrectly took ( F_2 )'s y-coordinate as 9/32 and divided by ( V_1V_2 ) which is 1/4, getting 9/8. But actually, the distance between F1 and F2 is 7/32, and ( V_1V_2 ) is 1/4, so the ratio is 7/8.Wait, so the user made a mistake in their calculation. They took ( F_2 )'s y-coordinate as 9/32 and divided by ( V_1V_2 )'s y-coordinate 1/4, but actually, both are distances along the y-axis, so ( F_1F_2 ) is 7/32 and ( V_1V_2 ) is 8/32, so the ratio is 7/8.Therefore, the correct ratio is 7/8, not 9/8.Wait, but let me double-check my own calculations.Given ( mathcal{Q} ) is ( y = 8x² + 1/4 ). So, vertex ( V_2 ) is at (0,1/4). Focus ( F_2 ) is at (0,1/4 + 1/(4*8)) = (0,1/4 + 1/32) = (0,8/32 + 1/32) = (0,9/32).( F_1 ) is at (0,1/16) = (0,2/32). So, the distance between ( F_1 ) and ( F_2 ) is |9/32 - 2/32| = 7/32.Distance between ( V_1 ) and ( V_2 ) is |1/4 - 0| = 1/4 = 8/32.Therefore, ratio ( frac{F_1F_2}{V_1V_2} = frac{7/32}{8/32} = 7/8 ).Yes, that's correct. So, the user's initial solution had an error in calculating the distance between ( F_1 ) and ( F_2 ). They took ( F_2 )'s y-coordinate as 9/32 and divided by ( V_1V_2 )'s y-coordinate 1/4, which is incorrect because ( F_1F_2 ) is the distance between the two foci, not just ( F_2 )'s coordinate.Therefore, the correct ratio is 7/8.Wait, but hold on. Let me check the initial problem again.Wait, the user wrote:"The focus ( F_2 ) (using ( a = 8 ) in the formula for the focus of ( y = ax^2 ) at ( (0, frac{1}{4a}) )) is at ( (0, frac{1}{4} + frac{1}{32}) )."Wait, hold on. The formula for the focus of ( y = ax² + c ) is ( (0, c + 1/(4a)) ). So, in this case, c is 1/4, and 1/(4a) is 1/32, so yes, ( F_2 ) is at 1/4 + 1/32 = 9/32.But when calculating the distance between ( F_1 ) and ( F_2 ), the user wrote:"The focus ( F_1 ) of ( mathcal{P} ) is at ( (0, frac{1}{16}) ), using formula ( F = (0, frac{1}{4a}) ) for ( y = 4x^2 )."Then, they calculated:( frac{F_1F_2}{V_1V_2} = frac{frac{9}{32}}{frac{1}{4}} = frac{9}{32} cdot 4 = boxed{frac{9}{8}} ).But this is incorrect because ( F_1F_2 ) is not 9/32; it's the distance between ( F_1 ) at 2/32 and ( F_2 ) at 9/32, which is 7/32.Therefore, the correct ratio is 7/8.So, in conclusion, the user's initial solution had an error in calculating the distance between the foci, leading to an incorrect ratio. The correct ratio is 7/8.But wait, let me make sure I didn't make any mistakes in my own reasoning.Let me recap:1. Given parabola ( y = 4x² ), vertex ( V_1 ) at (0,0), focus ( F_1 ) at (0,1/16).2. Points A(a,4a²) and B(b,4b²) on the parabola such that ( angle AV_1B = 90^circ ). This leads to ab = -1/16.3. Midpoint M of AB is ( ( (a+b)/2, 2(a² + b²) ) ).4. Using ( a² + b² = (a + b)² - 2ab ), and since ab = -1/16, we get ( a² + b² = (a + b)² + 1/8 ).5. Substituting into M's coordinates, we get ( (h, 8h² + 1/4) ), so the locus ( mathcal{Q} ) is ( y = 8x² + 1/4 ).6. Vertex ( V_2 ) of ( mathcal{Q} ) is at (0,1/4), focus ( F_2 ) is at (0,1/4 + 1/(4*8)) = (0,9/32).7. Distance ( F_1F_2 ) is |9/32 - 2/32| = 7/32.8. Distance ( V_1V_2 ) is |1/4 - 0| = 1/4 = 8/32.9. Ratio ( frac{F_1F_2}{V_1V_2} = frac{7/32}{8/32} = 7/8 ).Yes, that seems correct. So, the correct ratio is 7/8.Therefore, the user's initial solution had an error in calculating the ratio, leading to 9/8 instead of the correct 7/8.**Final Answer**boxed{dfrac{7}{8}}