💡Alright, so I need to figure out the domain of this function: y equals 1 over the square root of log base 0.5 of (4x minus 3). Hmm, okay, let's break this down step by step.First, I know that for a function to be defined, especially when there's a square root involved, the expression inside the square root has to be positive. Also, since it's in the denominator, the whole denominator can't be zero because division by zero is undefined. So, I need to make sure that the square root part is not only positive but also that the entire denominator isn't zero.Let me write down the function again to visualize it better: y = 1 / sqrt(log_{0.5}(4x - 3)). So, the denominator is sqrt(log_{0.5}(4x - 3)). For this to be defined, log_{0.5}(4x - 3) must be greater than zero because the square root of a negative number isn't real, and we can't have zero inside the square root either since that would make the denominator zero.Alright, so first condition: log_{0.5}(4x - 3) > 0.Now, I remember that logarithms have certain properties depending on their base. The base here is 0.5, which is between 0 and 1. I think that when the base of a logarithm is between 0 and 1, the logarithm function is decreasing. That means as the input increases, the output decreases, and vice versa.So, log_{0.5}(something) > 0. What does that tell me about "something"? Since the base is less than 1, log_{0.5}(a) > 0 implies that a < 1. Because, for example, log_{0.5}(1) is 0, and as the input decreases below 1, the logarithm becomes positive. Wait, let me check that.If I take log_{0.5}(0.5), that should be 1 because 0.5^1 = 0.5. And log_{0.5}(0.25) would be 2 because 0.5^2 = 0.25. So, as the input gets smaller, the logarithm increases. So, if log_{0.5}(4x - 3) > 0, then 4x - 3 must be less than 1 because log_{0.5}(1) is 0, and to get a positive value, the argument has to be less than 1.So, 4x - 3 < 1. Let me solve that inequality:4x - 3 < 1Add 3 to both sides:4x < 4Divide both sides by 4:x < 1Okay, so x has to be less than 1. But wait, I also need to make sure that the argument of the logarithm is positive because you can't take the logarithm of a non-positive number. So, 4x - 3 must be greater than 0.So, 4x - 3 > 0Add 3 to both sides:4x > 3Divide both sides by 4:x > 3/4So, x has to be greater than 3/4 and less than 1. Combining these two inequalities, we get 3/4 < x < 1.Let me just double-check to make sure I didn't make any mistakes. The logarithm base 0.5 of (4x - 3) has to be positive, which means 4x - 3 has to be less than 1, leading to x < 1. Also, 4x - 3 has to be positive, so x > 3/4. Therefore, the domain should be all real numbers between 3/4 and 1, not including the endpoints because at x = 3/4, 4x - 3 = 0, which is not allowed in the logarithm, and at x = 1, 4x - 3 = 1, which would make the logarithm zero, leading to division by zero in the original function.So, yeah, I think that's correct. The domain is all x such that 3/4 < x < 1.

💡Alright, so I'm trying to figure out this clock problem. It says that we need to consider all moments when the hour and minute hands of a clock lie on a straight line, forming a straight angle. Then, the question is asking if there can be any two such instances where the lines formed by these instances are mutually perpendicular.Okay, first, I need to understand what it means for the hour and minute hands to lie on a straight line. That means they're either overlapping (which would be a straight line in the same direction) or opposite each other (which would be a straight line in opposite directions). So, these are the two cases: when the hands overlap and when they are directly opposite.Now, the problem is asking if there can be two such instances where the lines formed by these straight angles are mutually perpendicular. So, essentially, we're looking for two different times when the hands are either overlapping or opposite, and the lines formed by these positions are perpendicular to each other.Let me think about how the angles of the hour and minute hands change over time. The minute hand moves 360 degrees in 60 minutes, which is 6 degrees per minute. The hour hand moves 360 degrees in 12 hours, which is 30 degrees per hour, or 0.5 degrees per minute.So, at any given time, the angle of the minute hand from 12 o'clock is 6 times the number of minutes, and the angle of the hour hand is 30 times the number of hours plus 0.5 times the number of minutes.If the hands are overlapping, their angles are equal. If they're opposite, their angles differ by 180 degrees.So, for overlapping:6m = 30h + 0.5mAnd for opposite:6m = 30h + 0.5m + 180Let me solve these equations to find the times when the hands are overlapping or opposite.Starting with overlapping:6m = 30h + 0.5m6m - 0.5m = 30h5.5m = 30hm = (30h)/5.5m = (60h)/11So, the minute hand overlaps the hour hand at m = (60h)/11 minutes past h o'clock.Similarly, for opposite:6m = 30h + 0.5m + 1806m - 0.5m = 30h + 1805.5m = 30h + 180m = (30h + 180)/5.5m = (60h + 360)/11So, the minute hand is opposite the hour hand at m = (60h + 360)/11 minutes past h o'clock.Now, these are the general solutions for overlapping and opposite positions. But the problem is asking about two such instances where the lines formed by these positions are mutually perpendicular.Hmm, so if we have two times when the hands are either overlapping or opposite, and the lines formed by these positions are perpendicular, what does that mean?I think it means that the angle between the two lines formed by the hands at these two different times is 90 degrees.So, let's say at time t1, the hands are either overlapping or opposite, forming a straight line at angle θ1. At time t2, the hands are either overlapping or opposite, forming another straight line at angle θ2. We need to check if θ2 - θ1 = 90 degrees or θ1 - θ2 = 90 degrees.But θ1 and θ2 are the angles of the hands at those times. So, θ1 = 6m1 (or 30h1 + 0.5m1, but since they're overlapping or opposite, θ1 = 6m1 or θ1 = 6m1 + 180).Similarly, θ2 = 6m2 or θ2 = 6m2 + 180.So, the difference between θ2 and θ1 should be 90 degrees.But θ1 and θ2 are both functions of time, so we need to find two times t1 and t2 such that |θ2 - θ1| = 90 degrees.But θ1 and θ2 are determined by the positions of the hands at those times, which are either overlapping or opposite.So, let's consider two cases:Case 1: Both t1 and t2 are overlapping times.Case 2: Both t1 and t2 are opposite times.Case 3: One is an overlapping time and the other is an opposite time.I think we need to check all these cases.Let's start with Case 1: Both t1 and t2 are overlapping times.At overlapping times, θ1 = 6m1 and θ2 = 6m2.We need |6m2 - 6m1| = 90 degrees.So, |m2 - m1| = 15 minutes.But m1 and m2 are given by m = (60h)/11.So, m2 - m1 = (60h2)/11 - (60h1)/11 = (60/11)(h2 - h1).We need this difference to be 15 minutes.So, (60/11)(h2 - h1) = 15Multiply both sides by 11:60(h2 - h1) = 165Divide both sides by 15:4(h2 - h1) = 11So, h2 - h1 = 11/4 = 2.75 hours.But h1 and h2 are integers (hours), so h2 - h1 must be an integer. But 2.75 is not an integer, so there's no solution in this case.Therefore, there are no two overlapping times where the lines formed are perpendicular.Now, let's check Case 2: Both t1 and t2 are opposite times.At opposite times, θ1 = 6m1 + 180 and θ2 = 6m2 + 180.We need |θ2 - θ1| = 90 degrees.So, |6m2 + 180 - (6m1 + 180)| = |6(m2 - m1)| = 90Thus, |m2 - m1| = 15 minutes.Again, m1 and m2 are given by m = (60h + 360)/11.So, m2 - m1 = (60h2 + 360)/11 - (60h1 + 360)/11 = (60/11)(h2 - h1).We need this difference to be 15 minutes.So, (60/11)(h2 - h1) = 15Multiply both sides by 11:60(h2 - h1) = 165Divide both sides by 15:4(h2 - h1) = 11Again, h2 - h1 = 11/4 = 2.75 hours, which is not an integer. So, no solution in this case either.Now, let's check Case 3: One time is overlapping, and the other is opposite.So, θ1 = 6m1 (overlapping) and θ2 = 6m2 + 180 (opposite).We need |θ2 - θ1| = 90 degrees.So, |6m2 + 180 - 6m1| = 90Simplify:|6(m2 - m1) + 180| = 90This gives two possibilities:6(m2 - m1) + 180 = 90 or 6(m2 - m1) + 180 = -90First equation:6(m2 - m1) + 180 = 906(m2 - m1) = -90m2 - m1 = -15Second equation:6(m2 - m1) + 180 = -906(m2 - m1) = -270m2 - m1 = -45So, m2 - m1 = -15 or m2 - m1 = -45But m1 is the overlapping time: m1 = (60h1)/11m2 is the opposite time: m2 = (60h2 + 360)/11So, m2 - m1 = (60h2 + 360)/11 - (60h1)/11 = (60(h2 - h1) + 360)/11We need this to be -15 or -45.First, let's set it equal to -15:(60(h2 - h1) + 360)/11 = -15Multiply both sides by 11:60(h2 - h1) + 360 = -16560(h2 - h1) = -525h2 - h1 = -525/60 = -8.75But h2 and h1 are integers, so h2 - h1 must be an integer. -8.75 is not an integer, so no solution.Now, set it equal to -45:(60(h2 - h1) + 360)/11 = -45Multiply both sides by 11:60(h2 - h1) + 360 = -49560(h2 - h1) = -855h2 - h1 = -855/60 = -14.25Again, not an integer, so no solution.Therefore, in all cases, there are no two instances where the lines formed by the hour and minute hands being on a straight line are mutually perpendicular.So, the answer is no, there cannot be any two such instances where the lines formed are mutually perpendicular.

💡Okay, so I have this problem here about a recursive sequence and a series that needs to be expressed in a specific form. Let me try to unpack it step by step.First, the problem introduces α as the inverse cosine of 2/3. So, α = cos⁻¹(2/3). That means cos(α) = 2/3. I remember that for such angles, sometimes it's useful to use trigonometric identities or maybe even complex numbers, but I'm not sure yet.Then, there's this recursive sequence a₀, a₁, a₂, and so on. It starts with a₀ = 1, which is straightforward. For each positive integer n, the term aₙ is given by this formula:aₙ = [cos(nα) - (a₁a_{n-1} + a₂a_{n-2} + ... + a_{n-1}a₁)] / (2a₀)Since a₀ is 1, this simplifies to:aₙ = cos(nα) - (a₁a_{n-1} + a₂a_{n-2} + ... + a_{n-1}a₁)Hmm, this looks like a recursive formula where each term aₙ is defined in terms of the previous terms. It reminds me of the kind of recursions that come up in generating functions, especially because of the convolution-like sum (a₁a_{n-1} + a₂a_{n-2} + ... + a_{n-1}a₁). That's similar to the coefficients of a product of two power series.So, maybe I should consider using generating functions here. Let me define the generating function A(x) as:A(x) = a₀ + a₁x + a₂x² + a₃x³ + ... Given that a₀ = 1, so A(x) starts with 1.Now, if I square A(x), I get:A(x)² = (a₀ + a₁x + a₂x² + a₃x³ + ...) * (a₀ + a₁x + a₂x² + a₃x³ + ...)Multiplying these out, the coefficient of xⁿ in A(x)² is exactly the sum (a₀aₙ + a₁a_{n-1} + a₂a_{n-2} + ... + aₙa₀). But in our recursion, we have (a₁a_{n-1} + a₂a_{n-2} + ... + a_{n-1}a₁), which is similar but missing the a₀aₙ term.Wait, so if I subtract a₀aₙ from both sides, I get:A(x)² - a₀A(x) = sum_{n=1}^∞ (a₁a_{n-1} + a₂a_{n-2} + ... + a_{n-1}a₁) xⁿBut in our recursion, aₙ = cos(nα) - (a₁a_{n-1} + ... + a_{n-1}a₁). So, rearranging, we get:a₁a_{n-1} + ... + a_{n-1}a₁ = cos(nα) - aₙTherefore, substituting back into the generating function equation:A(x)² - a₀A(x) = sum_{n=1}^∞ [cos(nα) - aₙ] xⁿBut wait, the left side is A(x)² - a₀A(x), and the right side is sum_{n=1}^∞ cos(nα) xⁿ - sum_{n=1}^∞ aₙ xⁿ.Notice that sum_{n=1}^∞ aₙ xⁿ is just A(x) - a₀, which is A(x) - 1.So, putting it all together:A(x)² - a₀A(x) = sum_{n=1}^∞ cos(nα) xⁿ - (A(x) - 1)But a₀ is 1, so:A(x)² - A(x) = sum_{n=1}^∞ cos(nα) xⁿ - A(x) + 1Simplify the left side:A(x)² - A(x) + A(x) - 1 = sum_{n=1}^∞ cos(nα) xⁿSo, A(x)² - 1 = sum_{n=1}^∞ cos(nα) xⁿBut the sum on the right is from n=1 to infinity, so let's write it as:sum_{n=1}^∞ cos(nα) xⁿ = sum_{n=0}^∞ cos(nα) xⁿ - cos(0α)Since cos(0α) = 1, we have:sum_{n=1}^∞ cos(nα) xⁿ = sum_{n=0}^∞ cos(nα) xⁿ - 1So, substituting back:A(x)² - 1 = sum_{n=0}^∞ cos(nα) xⁿ - 1Therefore, A(x)² = sum_{n=0}^∞ cos(nα) xⁿSo, now I need to find an expression for sum_{n=0}^∞ cos(nα) xⁿ.I remember that the sum of cos(nθ) xⁿ from n=0 to infinity is a known generating function. It can be expressed using complex exponentials.Recall that cos(nθ) = Re(e^{inθ}), so:sum_{n=0}^∞ cos(nθ) xⁿ = Re(sum_{n=0}^∞ (x e^{iθ})ⁿ) = Re(1 / (1 - x e^{iθ}))Similarly, the imaginary part would give the sum of sin(nθ) xⁿ.So, let's compute this:sum_{n=0}^∞ cos(nα) xⁿ = Re(1 / (1 - x e^{iα}))Let me compute this real part.First, write 1 / (1 - x e^{iα}) as:1 / (1 - x (cosα + i sinα)) = [1 - x cosα + i x sinα] / [(1 - x cosα)^2 + (x sinα)^2]Wait, how did I get that? Let me explain.If I have a complex denominator 1 - x e^{iα}, which is 1 - x cosα - i x sinα.To find the real part of 1 / (1 - x e^{iα}), I can multiply numerator and denominator by the complex conjugate of the denominator.So, the complex conjugate of 1 - x e^{iα} is 1 - x e^{-iα} = 1 - x cosα + i x sinα.So, multiplying numerator and denominator by this:[1 * (1 - x cosα + i x sinα)] / [(1 - x cosα - i x sinα)(1 - x cosα + i x sinα)]The denominator becomes:(1 - x cosα)^2 + (x sinα)^2Expanding this:1 - 2x cosα + x² cos²α + x² sin²αSince cos²α + sin²α = 1, this simplifies to:1 - 2x cosα + x²So, the denominator is 1 - 2x cosα + x².The numerator is 1 - x cosα + i x sinα.Therefore, 1 / (1 - x e^{iα}) = (1 - x cosα + i x sinα) / (1 - 2x cosα + x²)Taking the real part, we get:Re(1 / (1 - x e^{iα})) = (1 - x cosα) / (1 - 2x cosα + x²)Therefore, sum_{n=0}^∞ cos(nα) xⁿ = (1 - x cosα) / (1 - 2x cosα + x²)So, going back to our earlier equation, A(x)² = sum_{n=0}^∞ cos(nα) xⁿ = (1 - x cosα) / (1 - 2x cosα + x²)Therefore, A(x)² = (1 - x cosα) / (1 - 2x cosα + x²)So, A(x) = sqrt[(1 - x cosα) / (1 - 2x cosα + x²)]But we need to find the series sum_{k=0}^∞ a_k / 2^k, which is A(1/2).So, let's compute A(1/2):A(1/2) = sqrt[(1 - (1/2) cosα) / (1 - 2*(1/2) cosα + (1/2)^2)]Simplify numerator and denominator:Numerator: 1 - (1/2) cosαDenominator: 1 - cosα + 1/4So, let's compute this:First, cosα = 2/3, given that α = cos⁻¹(2/3).So, cosα = 2/3.Compute numerator:1 - (1/2)(2/3) = 1 - (1/3) = 2/3Compute denominator:1 - (2/3) + 1/4 = (1 - 2/3) + 1/4 = (1/3) + (1/4) = (4/12 + 3/12) = 7/12Therefore, A(1/2) = sqrt[(2/3) / (7/12)] = sqrt[(2/3) * (12/7)] = sqrt[(24/21)] = sqrt[(8/7)] = 2 sqrt(14) / 7Wait, let me check that:(2/3) divided by (7/12) is (2/3) * (12/7) = (24/21) = 8/7.So, sqrt(8/7) = 2 sqrt(14)/7? Wait, sqrt(8/7) is 2 sqrt(14)/7?Wait, sqrt(8/7) is sqrt(8)/sqrt(7) = 2 sqrt(2)/sqrt(7). Rationalizing the denominator, that's 2 sqrt(14)/7.Yes, that's correct.So, A(1/2) = 2 sqrt(14)/7.Therefore, the sum sum_{k=0}^∞ a_k / 2^k is equal to 2 sqrt(14)/7.Expressed as p sqrt(q)/r, where p and r are coprime positive integers, and q is square-free.Here, p = 2, q = 14, r = 7.Check if p and r are coprime: gcd(2,7) = 1, so yes.Check if q is square-free: 14 factors into 2 * 7, both primes, neither squared, so yes.Therefore, p + q + r = 2 + 14 + 7 = 23.So, the answer is 23.**Final Answer**boxed{23}

💡Okay, so I have to find the equation of a plane that contains three given points: (-2, 3, -1), (2, 5, 1), and (4, 1, 0). Hmm, I remember that to find the equation of a plane, I need a point on the plane and a normal vector to the plane. Since I have three points, I can use two of them to form vectors that lie on the plane, and then take the cross product of those vectors to get the normal vector. Once I have the normal vector, I can plug it into the plane equation formula along with one of the points to find the constant term. Let me try to break this down step by step.First, let me label the points for clarity. Let’s call the first point A(-2, 3, -1), the second point B(2, 5, 1), and the third point C(4, 1, 0). Now, I need to find two vectors that lie on the plane. I can do this by subtracting the coordinates of point A from points B and C. So, vector AB would be B - A, which is (2 - (-2), 5 - 3, 1 - (-1)) = (4, 2, 2). Similarly, vector AC would be C - A, which is (4 - (-2), 1 - 3, 0 - (-1)) = (6, -2, 1). Okay, so now I have two vectors on the plane: AB = (4, 2, 2) and AC = (6, -2, 1).Next, I need to find the normal vector to the plane. The normal vector can be found by taking the cross product of vectors AB and AC. I remember the cross product formula for two vectors (a1, a2, a3) and (b1, b2, b3) is given by:(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)So, applying this to vectors AB and AC:First component: (2)(1) - (2)(-2) = 2 + 4 = 6Second component: (2)(6) - (4)(1) = 12 - 4 = 8Wait, hold on, that doesn't seem right. Let me double-check. The cross product formula is (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1). So for the second component, it's a3b1 - a1b3. So, plugging in the values:a3 is 2, b1 is 6; a1 is 4, b3 is 1. So, 2*6 - 4*1 = 12 - 4 = 8. Hmm, okay, that seems correct.Third component: a1b2 - a2b1. So, a1 is 4, b2 is -2; a2 is 2, b1 is 6. So, 4*(-2) - 2*6 = -8 - 12 = -20.So, putting it all together, the cross product of AB and AC is (6, 8, -20). Therefore, the normal vector n is (6, 8, -20). Wait, but I think I made a mistake in the second component earlier. Let me verify again. The cross product formula is:If vector AB = (a1, a2, a3) = (4, 2, 2)and vector AC = (b1, b2, b3) = (6, -2, 1),then cross product n = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)So,First component: a2b3 - a3b2 = (2)(1) - (2)(-2) = 2 + 4 = 6Second component: a3b1 - a1b3 = (2)(6) - (4)(1) = 12 - 4 = 8Third component: a1b2 - a2b1 = (4)(-2) - (2)(6) = -8 - 12 = -20So, n = (6, 8, -20). Okay, that seems correct.But wait, the problem says to present the equation in the form Ax + By + Cz + D = 0, where A, B, C, D are integers with A > 0 and gcd(|A|, |B|, |C|, |D|) = 1. So, I think it's better if I simplify the normal vector by dividing by the greatest common divisor of its components.Looking at the normal vector (6, 8, -20). Let's find the gcd of 6, 8, and 20. The gcd of 6 and 8 is 2, and gcd of 2 and 20 is 2. So, I can divide each component by 2 to get a simpler normal vector.Dividing each component by 2: (6/2, 8/2, -20/2) = (3, 4, -10). So, the simplified normal vector is (3, 4, -10). Wait, but in the cross product, I got (6, 8, -20), which simplifies to (3, 4, -10). So, that's the normal vector. Now, I can use this normal vector to write the equation of the plane.The general equation of a plane is A(x - x0) + B(y - y0) + C(z - z0) = 0, where (x0, y0, z0) is a point on the plane, and (A, B, C) is the normal vector.I can choose any of the three given points to plug into the equation. Let me choose point A(-2, 3, -1) because it has smaller numbers, which might make calculations easier.So, plugging into the equation:3(x - (-2)) + 4(y - 3) - 10(z - (-1)) = 0Simplify each term:3(x + 2) + 4(y - 3) - 10(z + 1) = 0Now, distribute the constants:3x + 6 + 4y - 12 - 10z - 10 = 0Combine like terms:3x + 4y - 10z + (6 - 12 - 10) = 0Calculate the constants:6 - 12 = -6; -6 - 10 = -16So, the equation becomes:3x + 4y - 10z - 16 = 0Wait, but I think I made a mistake in the constants. Let me check again.After distributing:3x + 6 + 4y - 12 - 10z - 10 = 0Combine constants: 6 - 12 - 10 = 6 - 22 = -16So, 3x + 4y - 10z - 16 = 0. Hmm, but let me verify this equation with the other points to make sure it's correct.Let me plug in point B(2, 5, 1):3(2) + 4(5) - 10(1) - 16 = 6 + 20 - 10 - 16 = (6 + 20) + (-10 -16) = 26 - 26 = 0. Okay, that works.Now, let me plug in point C(4, 1, 0):3(4) + 4(1) - 10(0) - 16 = 12 + 4 - 0 - 16 = 16 - 16 = 0. That also works.Wait, so the equation 3x + 4y - 10z - 16 = 0 seems to satisfy all three points. But let me check my cross product again because earlier I thought I might have made a mistake.Wait, when I calculated the cross product, I got (6, 8, -20), which simplifies to (3, 4, -10). So, that seems correct. Then, using point A(-2, 3, -1), the equation becomes 3x + 4y - 10z - 16 = 0. But wait, in my initial calculation, I think I might have miscalculated the constants. Let me go through that step again.Starting with the normal vector (3, 4, -10) and point A(-2, 3, -1):3(x + 2) + 4(y - 3) - 10(z + 1) = 0Expanding:3x + 6 + 4y - 12 - 10z - 10 = 0Combine constants: 6 - 12 - 10 = -16So, 3x + 4y - 10z - 16 = 0. Yes, that's correct.Wait, but in the initial problem statement, the user presented an equation with D = -19, but in my calculation, I got D = -16. Hmm, maybe I made a mistake somewhere.Wait, let me check the cross product again. Maybe I messed up the cross product calculation.Vector AB is (4, 2, 2), vector AC is (6, -2, 1).Cross product:i component: (2)(1) - (2)(-2) = 2 + 4 = 6j component: -( (4)(1) - (2)(6) ) = -(4 - 12) = -(-8) = 8Wait, hold on, I think I made a mistake in the cross product formula. The cross product is (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1). But actually, the j component is negative of (a1b3 - a3b1). Wait, no, let me recall the cross product formula correctly.The cross product of AB × AC is:|i j k||4 2 2||6 -2 1|Which is i*(2*1 - 2*(-2)) - j*(4*1 - 2*6) + k*(4*(-2) - 2*6)Calculating each component:i: 2*1 - 2*(-2) = 2 + 4 = 6j: -(4*1 - 2*6) = -(4 - 12) = -(-8) = 8k: 4*(-2) - 2*6 = -8 - 12 = -20So, cross product is (6, 8, -20). So, that's correct.Then, simplifying by dividing by 2: (3, 4, -10). So, that's correct.Then, using point A(-2, 3, -1):3(x + 2) + 4(y - 3) - 10(z + 1) = 0Which expands to 3x + 6 + 4y - 12 - 10z - 10 = 0Combine constants: 6 - 12 - 10 = -16So, 3x + 4y - 10z - 16 = 0Wait, but in the initial problem, the user had D = -19. Hmm, maybe I made a mistake in the cross product or in the calculation.Wait, let me try using a different point to see if I get the same equation.Let me use point B(2, 5, 1):3(x - 2) + 4(y - 5) - 10(z - 1) = 0Expanding:3x - 6 + 4y - 20 - 10z + 10 = 0Combine constants: -6 - 20 + 10 = -16So, 3x + 4y - 10z - 16 = 0. Same result.Wait, so maybe the initial answer was incorrect? Because according to my calculations, it's 3x + 4y - 10z - 16 = 0, but the user had 3x + 5y - 10z - 19 = 0. Hmm, perhaps I made a mistake in the cross product.Wait, let me recalculate the cross product again.Vector AB = (4, 2, 2)Vector AC = (6, -2, 1)Cross product:i component: (2)(1) - (2)(-2) = 2 + 4 = 6j component: -( (4)(1) - (2)(6) ) = -(4 - 12) = -(-8) = 8k component: (4)(-2) - (2)(6) = -8 - 12 = -20So, cross product is (6, 8, -20). Simplify by dividing by 2: (3, 4, -10). So, that's correct.Wait, maybe the user made a mistake in their calculation. Because according to my calculations, the equation is 3x + 4y - 10z - 16 = 0, and it satisfies all three points.Wait, let me check the user's calculation. They had:Vectors AB = (4, 2, 2) and AC = (6, -2, 1)Cross product:( (2)(1) - (2)(-2), (2)(6) - (2)(1), (4)(-2) - (2)(6) ) = (6, 10, -20)Wait, hold on, that's different from what I got. The user got (6, 10, -20), but I got (6, 8, -20). So, that's the mistake. The user must have miscalculated the j component.Because the cross product formula is:i: a2b3 - a3b2j: a3b1 - a1b3k: a1b2 - a2b1So, for j component, it's a3b1 - a1b3.In the user's calculation, they did (2)(6) - (2)(1) = 12 - 2 = 10 for the j component. But according to the formula, it's a3b1 - a1b3, which is 2*6 - 4*1 = 12 - 4 = 8. So, the user incorrectly calculated the j component as 10 instead of 8. Therefore, their normal vector was (6, 10, -20), which simplifies to (3, 5, -10), leading to D = -19. But in reality, the correct cross product is (6, 8, -20), which simplifies to (3, 4, -10), leading to D = -16.So, the correct equation should be 3x + 4y - 10z - 16 = 0.Wait, but let me confirm this by plugging in all three points into both equations to see which one is correct.First, the user's equation: 3x + 5y - 10z - 19 = 0Testing point A(-2, 3, -1):3*(-2) + 5*3 - 10*(-1) - 19 = -6 + 15 + 10 - 19 = ( -6 + 15 ) + (10 - 19 ) = 9 - 9 = 0. Okay, that works.Point B(2, 5, 1):3*2 + 5*5 - 10*1 - 19 = 6 + 25 - 10 - 19 = (6 + 25) + (-10 -19) = 31 - 29 = 2 ≠ 0. So, that doesn't work. Therefore, the user's equation is incorrect.Now, my equation: 3x + 4y - 10z - 16 = 0Testing point A(-2, 3, -1):3*(-2) + 4*3 - 10*(-1) - 16 = -6 + 12 + 10 - 16 = ( -6 + 12 ) + (10 - 16 ) = 6 - 6 = 0. Correct.Point B(2, 5, 1):3*2 + 4*5 - 10*1 - 16 = 6 + 20 - 10 - 16 = (6 + 20) + (-10 -16) = 26 - 26 = 0. Correct.Point C(4, 1, 0):3*4 + 4*1 - 10*0 - 16 = 12 + 4 - 0 - 16 = 16 - 16 = 0. Correct.So, my equation works for all three points, while the user's equation only worked for point A and C but not B. Therefore, the correct equation is 3x + 4y - 10z - 16 = 0.Wait, but let me check the cross product again because the user got a different normal vector. Maybe I made a mistake in the cross product.Wait, cross product formula is:For vectors AB = (a1, a2, a3) and AC = (b1, b2, b3),n = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)So, plugging in:a1 = 4, a2 = 2, a3 = 2b1 = 6, b2 = -2, b3 = 1So,n_x = a2b3 - a3b2 = 2*1 - 2*(-2) = 2 + 4 = 6n_y = a3b1 - a1b3 = 2*6 - 4*1 = 12 - 4 = 8n_z = a1b2 - a2b1 = 4*(-2) - 2*6 = -8 - 12 = -20So, n = (6, 8, -20). So, that's correct.Therefore, the user made a mistake in calculating the j component as 10 instead of 8. So, their normal vector was incorrect, leading to an incorrect D value.So, the correct equation is 3x + 4y - 10z - 16 = 0.But wait, let me check if this equation can be simplified further. The coefficients are 3, 4, -10, and -16. The gcd of 3, 4, 10, and 16 is 1, so we can't simplify it further. Also, A = 3, which is positive, so that's good.Therefore, the equation of the plane is 3x + 4y - 10z - 16 = 0.Wait, but in the initial problem, the user had 3x + 5y - 10z - 19 = 0, which is different. So, I think the correct answer is 3x + 4y - 10z - 16 = 0.Wait, but let me double-check everything once more to be sure.Vectors AB = (4, 2, 2), AC = (6, -2, 1)Cross product n = (6, 8, -20) → simplified to (3, 4, -10)Using point A(-2, 3, -1):3(x + 2) + 4(y - 3) - 10(z + 1) = 0Expanding: 3x + 6 + 4y - 12 - 10z - 10 = 0 → 3x + 4y - 10z - 16 = 0Testing all points:A: 3*(-2) + 4*3 - 10*(-1) -16 = -6 + 12 +10 -16 = 0B: 3*2 + 4*5 -10*1 -16 = 6 +20 -10 -16 = 0C: 3*4 +4*1 -10*0 -16 =12 +4 -0 -16=0All correct.Therefore, the correct equation is 3x + 4y - 10z -16 =0.But wait, the user's answer was 3x +5y -10z -19=0, which doesn't satisfy point B. So, their answer is incorrect.So, in conclusion, the correct equation is 3x +4y -10z -16=0.

💡Okay, so I have this problem here about finding the equation of a circle. The center of the circle is given as (2, -1), and there's a chord intercepted by the line x - y - 1 = 0. The length of this chord is 2√2. Hmm, okay, let me try to visualize this.First, I remember that the equation of a circle is generally (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Since the center is (2, -1), the equation should start as (x - 2)^2 + (y + 1)^2 = r^2. So, I just need to find the radius r.Now, the chord is intercepted by the line x - y - 1 = 0, and its length is 2√2. I think I need to relate the length of the chord to the radius of the circle. I recall that the length of a chord can be related to the radius and the distance from the center of the circle to the chord. There's a formula for that, right?Let me recall. If d is the distance from the center to the chord, and l is the length of the chord, then the relationship is l = 2√(r^2 - d^2). So, if I can find d, the distance from the center (2, -1) to the line x - y - 1 = 0, I can plug in the values and solve for r.Okay, so how do I find the distance from a point to a line? I think the formula is |Ax + By + C| / √(A^2 + B^2), where Ax + By + C = 0 is the line and (x, y) is the point. Let me write that down.The line is x - y - 1 = 0. So, A = 1, B = -1, and C = -1. The point is (2, -1). Plugging into the formula: |1*2 + (-1)*(-1) + (-1)| / √(1^2 + (-1)^2). Let me compute the numerator first: 1*2 is 2, (-1)*(-1) is 1, and then + (-1) is -1. So, 2 + 1 - 1 equals 2. The absolute value of 2 is 2.Now, the denominator is √(1 + 1) which is √2. So, the distance d is 2 / √2. Simplifying that, 2 divided by √2 is √2. So, d = √2.Alright, so now I have the distance from the center to the chord, which is √2, and the length of the chord, which is 2√2. Let me plug these into the chord length formula.The formula is l = 2√(r^2 - d^2). So, substituting the known values: 2√2 = 2√(r^2 - (√2)^2). Let me simplify this equation.First, divide both sides by 2: √2 = √(r^2 - 2). Now, square both sides to eliminate the square roots: (√2)^2 = (r^2 - 2). That gives 2 = r^2 - 2. Then, add 2 to both sides: 4 = r^2. So, r^2 is 4, which means r is 2.Okay, so the radius is 2. Now, going back to the equation of the circle: (x - 2)^2 + (y + 1)^2 = r^2, which is 4. So, the equation is (x - 2)^2 + (y + 1)^2 = 4.Wait, let me double-check my steps to make sure I didn't make a mistake. First, the distance from the center to the line: I used the formula correctly, substituted the values, and got √2. Then, using the chord length formula, I set up the equation correctly, solved for r, and got 2. Plugging back into the circle equation, it all seems consistent.I think that's it. So, the equation of the circle is (x - 2)^2 + (y + 1)^2 = 4.**Final Answer**The equation of the circle is boxed{(x-2)^2 + (y+1)^2 = 4}.

💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand it step by step. The problem says:"Given an arithmetic sequence {a_n} with a non-zero common difference, a₁ = 1, and a₂, a₄, a₈ form a geometric sequence, let the sum of the first n terms of {a_n} be S_n. Then S_n equals to: [options given]"Alright, so first, let's recall what an arithmetic sequence is. An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. This constant is called the common difference, usually denoted by d. So, the nth term of an arithmetic sequence can be written as:a_n = a₁ + (n - 1)dGiven that a₁ = 1, so:a_n = 1 + (n - 1)dNow, the problem mentions that a₂, a₄, a₈ form a geometric sequence. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted by r.So, if a₂, a₄, a₈ are in geometric progression, then the ratio between a₄ and a₂ should be the same as the ratio between a₈ and a₄. In other words:a₄ / a₂ = a₈ / a₄Which implies that:(a₄)^2 = a₂ * a₈So, let's write expressions for a₂, a₄, and a₈ using the arithmetic sequence formula.a₂ = 1 + (2 - 1)d = 1 + da₄ = 1 + (4 - 1)d = 1 + 3da₈ = 1 + (8 - 1)d = 1 + 7dSo, substituting these into the equation:(1 + 3d)^2 = (1 + d)(1 + 7d)Let me expand both sides.Left side: (1 + 3d)^2 = 1^2 + 2*1*3d + (3d)^2 = 1 + 6d + 9d²Right side: (1 + d)(1 + 7d) = 1*1 + 1*7d + d*1 + d*7d = 1 + 7d + d + 7d² = 1 + 8d + 7d²So, setting both sides equal:1 + 6d + 9d² = 1 + 8d + 7d²Let me subtract 1 from both sides:6d + 9d² = 8d + 7d²Now, bring all terms to one side:6d + 9d² - 8d - 7d² = 0Simplify:(6d - 8d) + (9d² - 7d²) = 0-2d + 2d² = 0Factor out 2d:2d(-1 + d) = 0So, 2d = 0 or (-1 + d) = 0But the problem states that the common difference is non-zero, so d ≠ 0. Therefore, -1 + d = 0 implies d = 1.So, the common difference d is 1.Now, we need to find the sum of the first n terms, S_n.The formula for the sum of the first n terms of an arithmetic sequence is:S_n = n/2 * [2a₁ + (n - 1)d]We know a₁ = 1 and d = 1, so substituting:S_n = n/2 * [2*1 + (n - 1)*1] = n/2 * [2 + n - 1] = n/2 * (n + 1)So, S_n = n(n + 1)/2Looking at the options, that's option A.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I wrote the expressions for a₂, a₄, a₈ correctly.2. I set up the equation (1 + 3d)^2 = (1 + d)(1 + 7d) correctly.3. Expanded both sides correctly: left side is 1 + 6d + 9d², right side is 1 + 8d + 7d².4. Subtracted 1 from both sides: 6d + 9d² = 8d + 7d².5. Brought all terms to one side: 6d + 9d² - 8d - 7d² = 0, which simplifies to -2d + 2d² = 0.6. Factored out 2d: 2d(-1 + d) = 0, leading to d = 1.7. Calculated S_n using the formula, got n(n + 1)/2, which is option A.Everything seems correct. I don't see any mistakes in the calculations.**Final Answer**boxed{A}