💡Okay, so I have this problem where I need to find the largest possible distance between two complex numbers: (1 + 2i)z and z², given that |z| = 3. Hmm, complex numbers can sometimes be tricky, but let me try to break this down step by step.First, I know that the distance between two complex numbers a and b is given by the modulus |a - b|. So, in this case, I need to find the maximum value of |(1 + 2i)z - z²|. That makes sense.Since |z| = 3, z lies on a circle in the complex plane with radius 3 centered at the origin. I can represent z as 3e^{iθ}, where θ is the argument of z. That might make it easier to handle the modulus and operations on z.Let me write down the expression I need to maximize:|(1 + 2i)z - z²|Substituting z = 3e^{iθ} into the expression, I get:|(1 + 2i)(3e^{iθ}) - (3e^{iθ})²|Simplify each term:First term: (1 + 2i)(3e^{iθ}) = 3(1 + 2i)e^{iθ}Second term: (3e^{iθ})² = 9e^{i2θ}So, the expression becomes:|3(1 + 2i)e^{iθ} - 9e^{i2θ}|I can factor out a 3 from both terms:3| (1 + 2i)e^{iθ} - 3e^{i2θ} |So now, I need to maximize |(1 + 2i)e^{iθ} - 3e^{i2θ}| and then multiply by 3 to get the final distance.Let me denote A = (1 + 2i)e^{iθ} and B = 3e^{i2θ}. So, I need to find the maximum of |A - B|.Hmm, to find the maximum of |A - B|, I can think of A and B as vectors in the complex plane and find when they are in opposite directions, which would maximize their difference.But maybe it's better to express A and B in terms of their real and imaginary parts to compute |A - B|.Let me compute A first:A = (1 + 2i)e^{iθ} = (1 + 2i)(cosθ + i sinθ) = (cosθ - 2 sinθ) + i(2 cosθ + sinθ)Similarly, B = 3e^{i2θ} = 3(cos2θ + i sin2θ)So, A - B = [cosθ - 2 sinθ - 3 cos2θ] + i[2 cosθ + sinθ - 3 sin2θ]Therefore, |A - B| is the modulus of this complex number, which is:√[(cosθ - 2 sinθ - 3 cos2θ)² + (2 cosθ + sinθ - 3 sin2θ)²]This looks complicated, but maybe I can simplify it.Alternatively, perhaps using properties of complex numbers can help. Let me think.I know that |A - B| can be written as |A - B| = |A - B|, which is the same as |B - A|. So, maybe I can write it as |3e^{i2θ} - (1 + 2i)e^{iθ}|.Wait, that might not help directly. Maybe I can factor e^{iθ} out of both terms:|e^{iθ}(3e^{iθ} - (1 + 2i))|Since |e^{iθ}| = 1, this simplifies to |3e^{iθ} - (1 + 2i)|.So, now I have |3e^{iθ} - (1 + 2i)|, and I need to maximize this expression.That seems simpler. So, the problem reduces to finding the maximum of |3e^{iθ} - (1 + 2i)|.I remember that for a complex number w, the modulus |w - a| represents the distance from the point w to the point a in the complex plane. So, in this case, |3e^{iθ} - (1 + 2i)| is the distance from the point 3e^{iθ} (which lies on the circle of radius 3) to the fixed point (1 + 2i).To maximize this distance, I need to find the point on the circle |z| = 3 that is farthest from (1 + 2i). The maximum distance should be the distance from the origin to (1 + 2i) plus the radius of the circle, right?Wait, let me recall: if you have a circle centered at the origin with radius R, and a point a outside the circle, then the maximum distance from a to any point on the circle is |a| + R. Similarly, if a is inside the circle, the maximum distance would be R + |a| as well? Wait, no, if a is inside the circle, the maximum distance would still be R + |a|, but actually, if a is inside, the maximum distance is R + |a|, and the minimum is R - |a|.Wait, let me think again. If the circle is centered at the origin with radius R, and a is a point in the complex plane, then the maximum distance from a to any point on the circle is |a| + R, and the minimum is ||a| - R|.Yes, that's correct. So, in this case, the point (1 + 2i) has modulus |1 + 2i| = √(1² + 2²) = √5 ≈ 2.236, which is less than 3, so it's inside the circle of radius 3. Therefore, the maximum distance from (1 + 2i) to any point on the circle |z| = 3 is |1 + 2i| + 3 = √5 + 3 ≈ 5.236.Wait, but in our case, we have |3e^{iθ} - (1 + 2i)|. So, that's the distance from (1 + 2i) to the point 3e^{iθ} on the circle of radius 3. So, the maximum distance is indeed √5 + 3.But wait, hold on. Is that correct? Because 3e^{iθ} is on the circle of radius 3, and (1 + 2i) is inside that circle. So, the maximum distance between a point on the circle and the fixed point inside is indeed the distance from the center to the fixed point plus the radius.Yes, that seems right. So, the maximum of |3e^{iθ} - (1 + 2i)| is |1 + 2i| + 3 = √5 + 3.Therefore, going back to our original expression, we had:|(1 + 2i)z - z²| = 3|3e^{iθ} - (1 + 2i)|So, substituting the maximum value we found:3*(√5 + 3) = 3√5 + 9Wait, but hold on, earlier I thought the maximum distance was 12, but now I'm getting 3√5 + 9, which is approximately 3*2.236 + 9 ≈ 6.708 + 9 ≈ 15.708. That seems larger than 12. Did I make a mistake somewhere?Wait, let's go back. I think I messed up the factoring step. Let me double-check.We had:|(1 + 2i)z - z²| = |z| * |1 + 2i - z|Since |z| = 3, it becomes 3 * |1 + 2i - z|So, to maximize |1 + 2i - z|, where |z| = 3.So, it's the same as maximizing |z - (1 + 2i)|, which is the distance from z to (1 + 2i). Since z is on the circle |z| = 3, the maximum distance is |z| + |1 + 2i| = 3 + √5.Therefore, the maximum of |1 + 2i - z| is 3 + √5.Thus, the maximum of |(1 + 2i)z - z²| is 3*(3 + √5) = 9 + 3√5.Wait, but in the initial problem, the assistant's answer was 12, but according to this, it's 9 + 3√5, which is approximately 9 + 6.708 = 15.708.Hmm, that's a big discrepancy. So, which one is correct?Wait, maybe I made a mistake in the factoring step.Let me re-examine:|(1 + 2i)z - z²| = |z| * |1 + 2i - z|Is that correct?Wait, yes, because |ab| = |a||b|, so |(1 + 2i)z - z²| = |z| * |1 + 2i - z|.So, that step is correct.Then, since |z| = 3, it's 3 * |1 + 2i - z|.So, to maximize |1 + 2i - z|, which is the same as |z - (1 + 2i)|.Given that z is on the circle |z| = 3, the maximum distance from z to (1 + 2i) is |z| + |1 + 2i| = 3 + √5.Therefore, the maximum value is 3*(3 + √5) = 9 + 3√5.But wait, in the initial assistant's answer, they got 12. So, which one is correct?Wait, perhaps the assistant made a mistake in their reasoning. Let me see.In the assistant's solution, they said:| (1 + 2i)z - z² | = |z| * |1 + 2i - z| = 3 * |1 + 2i - z|Then, they said to maximize |1 + 2i - z|, assuming z = 3e^{iθ}, and then they set z = - (1 + 2i)/|1 + 2i| * 3, which is -3(1 + 2i)/√5.Then, they calculated |1 + 2i - z| as |1 + 2i + 3(1 + 2i)/√5| = (1 + 3/√5) * |1 + 2i| = (1 + 3/√5) * √5 = 4.Therefore, the maximum distance is 3 * 4 = 12.Wait, so according to this, the maximum of |1 + 2i - z| is 4, which would make the total distance 12.But according to my previous reasoning, the maximum should be 3 + √5, which is approximately 5.236, leading to 3*5.236 ≈ 15.708.So, which one is correct?Wait, perhaps the assistant made a mistake in their calculation.Let me compute |1 + 2i - z| when z = -3(1 + 2i)/√5.So, z = -3(1 + 2i)/√5.Then, 1 + 2i - z = 1 + 2i - (-3(1 + 2i)/√5) = 1 + 2i + 3(1 + 2i)/√5.Factor out (1 + 2i):(1 + 3/√5)(1 + 2i).Then, the modulus is |1 + 3/√5| * |1 + 2i|.|1 + 3/√5| is just 1 + 3/√5, since it's a positive real number.|1 + 2i| is √5.Therefore, the modulus is (1 + 3/√5)*√5 = √5 + 3.Which is exactly what I had earlier.So, the assistant's calculation seems to have an error. They said:|1 + 2i - z| = |1 + 2i + 3(1 + 2i)/√5| = (1 + 3/√5) * |1 + 2i| = (1 + 3/√5) * √5 = 4.But wait, (1 + 3/√5)*√5 = √5 + 3, not 4.Because (1 + 3/√5)*√5 = √5 + 3.So, the assistant incorrectly calculated this as 4, which is wrong.Therefore, the correct maximum of |1 + 2i - z| is √5 + 3, so the maximum distance is 3*(√5 + 3) = 9 + 3√5.But let me double-check this.Alternatively, perhaps the maximum occurs when z is in the opposite direction of (1 + 2i). Let me visualize this.The point (1 + 2i) is in the first quadrant, and the circle |z| = 3 is centered at the origin with radius 3. The farthest point from (1 + 2i) on this circle would be in the direction opposite to (1 + 2i), i.e., in the direction of -(1 + 2i).So, the point z would be -k*(1 + 2i), where k is a positive real number such that |z| = 3.So, |z| = | -k*(1 + 2i) | = k*|1 + 2i| = k*√5 = 3 => k = 3/√5.Therefore, z = -3(1 + 2i)/√5.Then, the distance from z to (1 + 2i) is |z - (1 + 2i)| = | -3(1 + 2i)/√5 - (1 + 2i) | = | (-3/√5 - 1)(1 + 2i) |.Factor out (1 + 2i):| (-3/√5 - 1) | * |1 + 2i| = (3/√5 + 1) * √5 = 3 + √5.So, that's consistent with what I had before.Therefore, the maximum of |1 + 2i - z| is indeed 3 + √5, so the maximum distance is 3*(3 + √5) = 9 + 3√5.But wait, let me compute 9 + 3√5 numerically to see what it is.√5 ≈ 2.236, so 3√5 ≈ 6.708, so 9 + 6.708 ≈ 15.708.But the assistant's answer was 12, which is less than 15.708. So, clearly, there's a mistake in the assistant's solution.Wait, perhaps I made a mistake in interpreting the problem. Let me read it again."Find the largest possible distance between (1 + 2i)z and z² when plotted in the complex plane."So, yes, it's |(1 + 2i)z - z²|.So, my reasoning seems correct, but the assistant's answer is 12, which is less than what I calculated.Wait, maybe the assistant made a mistake in their calculation when they said:|1 + 2i - z| = |1 + 2i + 3(1 + 2i)/√5| = (1 + 3/√5) * |1 + 2i| = 4.But as I showed, (1 + 3/√5)*√5 = √5 + 3, not 4.Wait, let me compute (1 + 3/√5)*√5:= √5 + 3.Yes, that's correct.So, the assistant incorrectly simplified it to 4, which is wrong.Therefore, the correct maximum distance is 3*(3 + √5) = 9 + 3√5.But let me see if there's another approach to confirm this.Alternatively, perhaps using calculus to maximize the expression.Let me express |(1 + 2i)z - z²| as |z| * |1 + 2i - z|, which is 3 * |1 + 2i - z|.So, to maximize |1 + 2i - z|, where |z| = 3.Let me write z = x + yi, where x² + y² = 9.Then, 1 + 2i - z = (1 - x) + (2 - y)i.So, |1 + 2i - z| = √[(1 - x)² + (2 - y)²]We need to maximize this expression subject to x² + y² = 9.This is a constrained optimization problem. We can use the method of Lagrange multipliers.Let me set up the function to maximize:f(x, y) = (1 - x)² + (2 - y)²Subject to the constraint:g(x, y) = x² + y² - 9 = 0The Lagrangian is:L(x, y, λ) = (1 - x)² + (2 - y)² - λ(x² + y² - 9)Taking partial derivatives:∂L/∂x = -2(1 - x) - 2λx = 0∂L/∂y = -2(2 - y) - 2λy = 0∂L/∂λ = -(x² + y² - 9) = 0So, from ∂L/∂x = 0:-2(1 - x) - 2λx = 0 => - (1 - x) - λx = 0 => -1 + x - λx = 0 => x(1 - λ) = 1 => x = 1 / (1 - λ)Similarly, from ∂L/∂y = 0:-2(2 - y) - 2λy = 0 => - (2 - y) - λy = 0 => -2 + y - λy = 0 => y(1 - λ) = 2 => y = 2 / (1 - λ)So, x = 1 / (1 - λ) and y = 2 / (1 - λ). Therefore, x/y = 1/2, so y = 2x.Now, using the constraint x² + y² = 9, and y = 2x:x² + (2x)² = 9 => x² + 4x² = 9 => 5x² = 9 => x² = 9/5 => x = ±3/√5Therefore, x = 3/√5 or x = -3/√5Similarly, y = 2x = 6/√5 or y = -6/√5So, we have two critical points:z1 = (3/√5) + (6/√5)iz2 = (-3/√5) + (-6/√5)iNow, let's compute |1 + 2i - z| for both points.First, for z1:1 + 2i - z1 = (1 - 3/√5) + (2 - 6/√5)iCompute the modulus:√[(1 - 3/√5)² + (2 - 6/√5)²]Let me compute each term:(1 - 3/√5)² = 1 - 6/√5 + 9/5 = (5 - 6√5 + 9)/5 = (14 - 6√5)/5(2 - 6/√5)² = 4 - 24/√5 + 36/5 = (20 - 24√5 + 36)/5 = (56 - 24√5)/5Adding them together:(14 - 6√5 + 56 - 24√5)/5 = (70 - 30√5)/5 = 14 - 6√5So, the modulus is √(14 - 6√5). Let me compute this numerically:√5 ≈ 2.236, so 6√5 ≈ 13.41614 - 13.416 ≈ 0.584√0.584 ≈ 0.764So, |1 + 2i - z1| ≈ 0.764Now, for z2:1 + 2i - z2 = (1 + 3/√5) + (2 + 6/√5)iCompute the modulus:√[(1 + 3/√5)² + (2 + 6/√5)²]Compute each term:(1 + 3/√5)² = 1 + 6/√5 + 9/5 = (5 + 6√5 + 9)/5 = (14 + 6√5)/5(2 + 6/√5)² = 4 + 24/√5 + 36/5 = (20 + 24√5 + 36)/5 = (56 + 24√5)/5Adding them together:(14 + 6√5 + 56 + 24√5)/5 = (70 + 30√5)/5 = 14 + 6√5So, the modulus is √(14 + 6√5). Let's compute this numerically:√5 ≈ 2.236, so 6√5 ≈ 13.41614 + 13.416 ≈ 27.416√27.416 ≈ 5.236So, |1 + 2i - z2| ≈ 5.236, which is exactly 3 + √5, since √5 ≈ 2.236, so 3 + 2.236 ≈ 5.236.Therefore, the maximum value of |1 + 2i - z| is indeed 3 + √5, and the minimum is √(14 - 6√5) ≈ 0.764.Therefore, the maximum distance is 3*(3 + √5) = 9 + 3√5.So, the assistant's answer of 12 is incorrect because they made a mistake in their calculation when they said:|1 + 2i - z| = |1 + 2i + 3(1 + 2i)/√5| = (1 + 3/√5) * |1 + 2i| = 4.But as we saw, this should be √5 + 3, not 4.Therefore, the correct maximum distance is 9 + 3√5.But let me compute 9 + 3√5 numerically to see what it is:√5 ≈ 2.236, so 3√5 ≈ 6.7089 + 6.708 ≈ 15.708So, the maximum distance is approximately 15.708.But the assistant's answer was 12, which is significantly less.Wait, perhaps the assistant made a mistake in the initial step when they factored out |z|.Let me re-examine their solution:They wrote:| (1 + 2i)z - z² | = |z| * |1 + 2i - z| = 3 * |1 + 2i - z|But wait, is that correct?Wait, |(1 + 2i)z - z²| = |z| * |1 + 2i - z|.Yes, that's correct because |ab - ac| = |a||b - c|.So, that step is correct.Then, they said:"Since |z| = 3, z lies on a circle with radius 3 centered at the origin in the complex plane. We want to maximize |1 + 2i - z|."Then, they assumed z = 3e^{iθ}, which is correct.Then, they said:"We maximize this expression when 1 + 2i and z = 3e^{iθ} are diametrically opposite, enhancing by radius 3. Hence, this is when z = - (1 + 2i)/|1 + 2i| * 3 = -3(1 + 2i)/√5."Then, they computed:|1 + 2i - z| = |1 + 2i + 3(1 + 2i)/√5| = |(1 + 3/√5)(1 + 2i)| = (1 + 3/√5) * √5 = 4.But as we saw, this should be √5 + 3, not 4.So, the assistant incorrectly calculated this step, leading to an incorrect final answer.Therefore, the correct maximum distance is 9 + 3√5.But let me see if there's another way to think about this.Alternatively, perhaps using the triangle inequality.We have |(1 + 2i)z - z²| = |z| * |1 + 2i - z| = 3 * |1 + 2i - z|.To maximize this, we need to maximize |1 + 2i - z|.Since z is on the circle |z| = 3, the maximum of |1 + 2i - z| is |1 + 2i| + |z| = √5 + 3, by the triangle inequality.Therefore, the maximum distance is 3*(√5 + 3) = 9 + 3√5.Yes, that's consistent with what I found earlier.Therefore, the correct answer should be 9 + 3√5, not 12.So, the assistant made a mistake in their calculation when they said the modulus was 4, when it should have been √5 + 3.Therefore, the largest possible distance is 9 + 3√5.But let me compute this exactly:9 + 3√5 = 3*(3 + √5)But perhaps we can write it as 3(3 + √5), but it's already simplified.Alternatively, if we rationalize or present it differently, but I think 9 + 3√5 is fine.So, to conclude, the largest possible distance between (1 + 2i)z and z² is 9 + 3√5.But wait, let me check if this is indeed the maximum.Alternatively, perhaps I can think of z² as a point on the circle of radius 9, since |z²| = |z|² = 9.And (1 + 2i)z is a point on the circle of radius 3*√5, since |(1 + 2i)z| = |1 + 2i|*|z| = √5*3 = 3√5.So, the distance between a point on the circle of radius 3√5 and a point on the circle of radius 9.The maximum distance between two points on these circles would be 3√5 + 9, which is the same as 9 + 3√5.Yes, that makes sense.Therefore, the maximum distance is indeed 9 + 3√5.So, the assistant's answer was incorrect because they made a mistake in their calculation, leading them to an incorrect result of 12.Therefore, the correct answer is 9 + 3√5.But wait, let me compute 9 + 3√5 numerically to see if it's indeed the maximum.As I did before, √5 ≈ 2.236, so 3√5 ≈ 6.708, so 9 + 6.708 ≈ 15.708.Is there a way to confirm this?Alternatively, perhaps using geometry.The point (1 + 2i) is inside the circle |z| = 3, so the maximum distance from (1 + 2i) to any point on the circle is 3 + √5, as we saw.Therefore, the maximum of |1 + 2i - z| is 3 + √5, so multiplying by 3 gives 9 + 3√5.Yes, that seems correct.Therefore, the largest possible distance is 9 + 3√5.So, the assistant's answer was wrong because they incorrectly calculated the modulus as 4 instead of √5 + 3.Therefore, the correct answer is 9 + 3√5.But wait, let me see if there's another perspective.Alternatively, perhaps using the reverse triangle inequality.We have | |a| - |b| | ≤ |a ± b| ≤ |a| + |b|.But in our case, we have |(1 + 2i)z - z²|.But I think the approach I took earlier is sufficient.Therefore, I'm confident that the correct answer is 9 + 3√5.

💡First, I need to find the values of ( x ) for which ( f(x) = frac{1}{4} ) given the piecewise function:[f(x) = begin{cases} 2^{-x}, & x in (-infty, 1] log_{81} x, & x in (1, +infty)end{cases}]I'll consider each piece of the function separately.**For ( x leq 1 ):**[2^{-x} = frac{1}{4}]Since ( frac{1}{4} = 2^{-2} ), I can set the exponents equal:[-x = -2 implies x = 2]However, ( x = 2 ) does not satisfy ( x leq 1 ), so this solution is discarded.**For ( x > 1 ):**[log_{81} x = frac{1}{4}]Rewriting the logarithmic equation in exponential form:[x = 81^{frac{1}{4}} = sqrt[4]{81} = 3]Since ( x = 3 ) satisfies ( x > 1 ), this is a valid solution.**Conclusion:**The only value of ( x ) that satisfies ( f(x) = frac{1}{4} ) is ( x = 3 ).

💡Alright, so I've got this problem here: "If the two sides of a right triangle are the two roots of the equation (x^{2} - 7x + 12 = 0), then the area of the right triangle is ( )." The options are A: 6, B: 12, C: 12 or (frac{3sqrt{7}}{2}), and D: 6 or (frac{3sqrt{7}}{2}).Okay, let's break this down step by step. First, I need to find the roots of the quadratic equation (x^{2} - 7x + 12 = 0). I remember that quadratic equations can often be factored, so let me try that.Looking at the equation (x^{2} - 7x + 12), I need two numbers that multiply to 12 and add up to -7. Hmm, 3 and 4 multiply to 12, and if both are negative, they add up to -7. So, the factors should be ((x - 3)(x - 4) = 0). Let me check that:[(x - 3)(x - 4) = x^{2} - 4x - 3x + 12 = x^{2} - 7x + 12]Yes, that's correct. So, the roots are (x = 3) and (x = 4). These are the lengths of two sides of a right triangle.Now, in a right triangle, the sides can be either both legs or one leg and the hypotenuse. So, I need to consider two scenarios:1. Both 3 and 4 are the legs of the triangle.2. One of them is the hypotenuse, and the other is a leg.Let's start with the first scenario. If both 3 and 4 are the legs, then the area of the triangle is straightforward:[text{Area} = frac{1}{2} times text{base} times text{height} = frac{1}{2} times 3 times 4 = frac{12}{2} = 6]So, the area is 6 in this case. That's one possibility.Now, the second scenario: one of the sides is the hypotenuse. Since 4 is larger than 3, let's assume 4 is the hypotenuse. Then, the other side (3) is a leg, and we need to find the other leg. Let's denote the unknown leg as (b). Using the Pythagorean theorem:[a^{2} + b^{2} = c^{2}]Where (a = 3), (c = 4), so:[3^{2} + b^{2} = 4^{2} 9 + b^{2} = 16 b^{2} = 16 - 9 = 7 b = sqrt{7}]So, the other leg is (sqrt{7}). Now, the area of the triangle would be:[text{Area} = frac{1}{2} times 3 times sqrt{7} = frac{3sqrt{7}}{2}]Therefore, in this configuration, the area is (frac{3sqrt{7}}{2}).Wait a second, could 3 be the hypotenuse instead? Let me check that. If 3 is the hypotenuse, then the other sides would have to satisfy:[a^{2} + b^{2} = 3^{2} = 9]But one of the sides is 4, which is larger than 3. That's impossible because the hypotenuse is the longest side in a right triangle. So, 3 cannot be the hypotenuse. Therefore, only 4 can be the hypotenuse in this case.So, summarizing the two scenarios:1. Both 3 and 4 are legs: Area = 62. 4 is the hypotenuse and 3 is a leg: Area = (frac{3sqrt{7}}{2})Therefore, the area of the right triangle can be either 6 or (frac{3sqrt{7}}{2}).Looking back at the options:A: 6B: 12C: 12 or (frac{3sqrt{7}}{2})D: 6 or (frac{3sqrt{7}}{2})So, the correct answer is D.Wait, just to make sure I didn't make any calculation mistakes. Let me verify the areas again.First scenario: legs 3 and 4.Area = 1/2 * 3 * 4 = 6. That seems right.Second scenario: hypotenuse 4, leg 3.Other leg = sqrt(4^2 - 3^2) = sqrt(16 - 9) = sqrt(7). So, area = 1/2 * 3 * sqrt(7) = (3 sqrt(7))/2. That also seems correct.Yes, I think I did everything right. So, the answer should be D.**Final Answer**boxed{D}

💡Okay, so I need to figure out the value of scalar ( k ) such that any points ( A ), ( B ), ( C ), and ( D ) satisfying the equation[ 4 overrightarrow{OA} - 3 overrightarrow{OB} + 7 overrightarrow{OC} + k overrightarrow{OD} = mathbf{0} ]are guaranteed to be coplanar. Hmm, coplanar means all four points lie on the same plane. I remember that for four points to be coplanar, the volume of the tetrahedron formed by them should be zero. But I'm not sure if that's the right approach here. Maybe there's another way.Let me think about vectors. The equation given is a linear combination of the position vectors of points ( A ), ( B ), ( C ), and ( D ) with respect to the origin ( O ). So, if I rearrange the equation, I can write it as:[ 4 overrightarrow{OA} - 3 overrightarrow{OB} + 7 overrightarrow{OC} = -k overrightarrow{OD} ]Or, dividing both sides by ( -k ):[ overrightarrow{OD} = -frac{4}{k} overrightarrow{OA} + frac{3}{k} overrightarrow{OB} - frac{7}{k} overrightarrow{OC} ]This looks like expressing ( overrightarrow{OD} ) as a linear combination of ( overrightarrow{OA} ), ( overrightarrow{OB} ), and ( overrightarrow{OC} ). For point ( D ) to be coplanar with ( A ), ( B ), and ( C ), the coefficients of this linear combination must add up to 1. That is, the sum of the coefficients should be 1. Let me check if that's a valid condition.So, the coefficients are ( -frac{4}{k} ), ( frac{3}{k} ), and ( -frac{7}{k} ). Adding them up:[ -frac{4}{k} + frac{3}{k} - frac{7}{k} = frac{-4 + 3 - 7}{k} = frac{-8}{k} ]For ( D ) to be coplanar with ( A ), ( B ), and ( C ), this sum should equal 1. So,[ frac{-8}{k} = 1 ]Solving for ( k ):[ -8 = k ]So, ( k = -8 ).Wait, let me verify this. If ( k = -8 ), then the equation becomes:[ 4 overrightarrow{OA} - 3 overrightarrow{OB} + 7 overrightarrow{OC} - 8 overrightarrow{OD} = mathbf{0} ]Which can be rewritten as:[ 4 overrightarrow{OA} - 3 overrightarrow{OB} + 7 overrightarrow{OC} = 8 overrightarrow{OD} ]Dividing both sides by 8:[ overrightarrow{OD} = frac{4}{8} overrightarrow{OA} - frac{3}{8} overrightarrow{OB} + frac{7}{8} overrightarrow{OC} ]Simplifying:[ overrightarrow{OD} = frac{1}{2} overrightarrow{OA} - frac{3}{8} overrightarrow{OB} + frac{7}{8} overrightarrow{OC} ]Now, adding up the coefficients:[ frac{1}{2} - frac{3}{8} + frac{7}{8} = frac{4}{8} - frac{3}{8} + frac{7}{8} = frac{8}{8} = 1 ]Yes, that checks out. So, when ( k = -8 ), point ( D ) lies in the plane formed by points ( A ), ( B ), and ( C ). Therefore, all four points are coplanar.I think that makes sense. The key was recognizing that for ( D ) to be coplanar with ( A ), ( B ), and ( C ), the coefficients in the linear combination must sum to 1. That's a standard condition in affine combinations, right? So, by ensuring that the sum of the coefficients equals 1, we guarantee that ( D ) lies in the plane defined by ( A ), ( B ), and ( C ).Just to make sure, let me think about another approach. Maybe using vectors and the scalar triple product. If four points are coplanar, the volume of the tetrahedron formed by them is zero, which can be expressed using the scalar triple product. The scalar triple product of vectors ( overrightarrow{AB} ), ( overrightarrow{AC} ), and ( overrightarrow{AD} ) should be zero.But in this case, the given equation is a linear combination of the position vectors. Maybe expressing the condition in terms of barycentric coordinates or something similar. But I think the approach I took earlier is sufficient and straightforward.So, to recap: The equation given is a linear combination of the position vectors. For point ( D ) to be coplanar with ( A ), ( B ), and ( C ), the coefficients in the linear combination must sum to 1. Setting up that equation gives ( k = -8 ). I verified this by plugging it back into the equation and checking the sum of coefficients. It works out, so I'm confident that ( k = -8 ) is the correct answer.**Final Answer**The value of ( k ) is boxed{-8}.

💡Okay, so I have this problem here about points on a parabola and their distances to the focus. Let me try to figure this out step by step. First, the parabola given is ( y^2 = 8x ). I remember that the standard form of a parabola that opens to the right is ( y^2 = 4ax ), where ( a ) is the distance from the vertex to the focus. Comparing this to the given equation, ( 4a = 8 ), so ( a = 2 ). That means the focus ( F ) of this parabola is at ( (2, 0) ).Now, the problem mentions four points ( P_1, P_2, P_3, ) and ( P_4 ) on this parabola, each with their respective ( x )-coordinates ( x_1, x_2, x_3, ) and ( x_4 ). It also tells me that the sum of these ( x )-coordinates is 10, so ( x_1 + x_2 + x_3 + x_4 = 10 ).I need to find the sum of the distances from each of these points to the focus ( F ). That is, I need to compute ( |P_1F| + |P_2F| + |P_3F| + |P_4F| ).I recall that for any point on a parabola, the distance from that point to the focus is equal to its distance to the directrix. For the parabola ( y^2 = 4ax ), the directrix is the line ( x = -a ). So, for our parabola ( y^2 = 8x ), the directrix is ( x = -2 ).But how does this help me find the distance from a point ( P ) on the parabola to the focus ( F )? Let me think. If I have a point ( P ) on the parabola with coordinates ( (x, y) ), then the distance from ( P ) to ( F ) can be calculated using the distance formula. The focus is at ( (2, 0) ), so the distance ( |PF| ) is:[|PF| = sqrt{(x - 2)^2 + y^2}]But since ( P ) lies on the parabola ( y^2 = 8x ), I can substitute ( y^2 ) with ( 8x ) in the distance formula:[|PF| = sqrt{(x - 2)^2 + 8x}]Let me simplify this expression. Expanding ( (x - 2)^2 ):[(x - 2)^2 = x^2 - 4x + 4]So, plugging that back into the distance formula:[|PF| = sqrt{x^2 - 4x + 4 + 8x} = sqrt{x^2 + 4x + 4}]Hmm, ( x^2 + 4x + 4 ) is a perfect square. It factors into ( (x + 2)^2 ). So,[|PF| = sqrt{(x + 2)^2} = |x + 2|]Since ( x ) is a coordinate on the parabola ( y^2 = 8x ), ( x ) must be non-negative because ( y^2 ) is always non-negative. Therefore, ( x + 2 ) is always positive, so the absolute value can be removed:[|PF| = x + 2]Oh, that's a neat result! So, for any point ( P ) on this parabola, the distance from ( P ) to the focus ( F ) is simply ( x + 2 ), where ( x ) is the ( x )-coordinate of ( P ).That makes the problem easier. Instead of dealing with square roots and complicated expressions, I can just use this linear relationship. So, for each point ( P_i ), the distance to ( F ) is ( x_i + 2 ).Therefore, the sum ( |P_1F| + |P_2F| + |P_3F| + |P_4F| ) is equal to:[(x_1 + 2) + (x_2 + 2) + (x_3 + 2) + (x_4 + 2)]Let me compute this:[x_1 + x_2 + x_3 + x_4 + 2 + 2 + 2 + 2 = (x_1 + x_2 + x_3 + x_4) + 8]But we are given that ( x_1 + x_2 + x_3 + x_4 = 10 ). So substituting that in:[10 + 8 = 18]So, the total distance is 18. Let me just double-check my steps to make sure I didn't make any mistakes.1. I identified the parabola equation and found the focus correctly.2. I used the distance formula and substituted ( y^2 ) with ( 8x ) correctly.3. I expanded and simplified the expression correctly, recognizing the perfect square.4. I correctly noted that ( x ) is non-negative, so the absolute value can be removed.5. I applied this result to each point and summed them up, correctly substituting the given sum of ( x )-coordinates.Everything seems to check out. So, I'm confident that the answer is 18.**Final Answer**boxed{B}

💡Okay, so I have this problem about constructing a sequence of isosceles triangles, each time using a base angle from the previous triangle as the vertex angle for the next one. The initial triangle, triangle ABC, is isosceles with a vertex angle of π/5. I need to find the first few angles α₂, α₃, α₄ and then figure out a general formula for αₙ.First, let me visualize the problem. Triangle ABC is isosceles with vertex angle at A, which is π/5. So, the two base angles at B and C must each be (π - π/5)/2. Let me calculate that:Base angle = (π - π/5)/2 = (4π/5)/2 = 2π/5.So, angles at B and C are each 2π/5. Now, for triangle 2, which is triangle BCD, we use one of the base angles of triangle 1 as the vertex angle. That would be angle ABC, which is 2π/5. So, triangle BCD is isosceles with vertex angle at B equal to 2π/5. Then, the base angles at C and D would each be (π - 2π/5)/2.Let me compute that:Base angle = (π - 2π/5)/2 = (3π/5)/2 = 3π/10.So, α₂ is 3π/10. Wait, hold on, is α₂ the vertex angle or the base angle? The problem says that α₂ is the vertex angle of triangle 2, which is angle DBC. So, actually, α₂ is 2π/5, not 3π/10. Hmm, maybe I got confused here.Wait, let me read the problem again. It says: "Construct an isosceles triangle triangle₂ (i.e., triangle BCD) with one of the base angles of triangle₁ (angle ABC) as the vertex angle, denoted as α₂ (α₂ = angle DBC)." So, angle DBC is the vertex angle of triangle BCD, which is equal to angle ABC from triangle ABC, which is 2π/5. So, α₂ is 2π/5.But then, the base angles of triangle BCD would be (π - α₂)/2 = (π - 2π/5)/2 = (3π/5)/2 = 3π/10. So, angles at C and D in triangle BCD are each 3π/10.Then, moving on to triangle 3, which is triangle CEB. We use one of the base angles of triangle 2 as the vertex angle. So, the base angles of triangle 2 are 3π/10, so α₃ is 3π/10. Wait, no, the vertex angle of triangle 3 is one of the base angles of triangle 2, which is 3π/10. So, α₃ = 3π/10.But then, the base angles of triangle 3 would be (π - α₃)/2 = (π - 3π/10)/2 = (7π/10)/2 = 7π/20.Wait, so α₃ is 3π/10, and the base angles are 7π/20. Then, for triangle 4, we use one of the base angles of triangle 3, which is 7π/20, as the vertex angle. So, α₄ = 7π/20.Let me summarize:- α₁ = π/5- α₂ = 2π/5- α₃ = 3π/10- α₄ = 7π/20Wait, that seems to follow a pattern. Let me check if this is correct.Starting with triangle 1: vertex angle α₁ = π/5, base angles = (π - π/5)/2 = 2π/5 each.Triangle 2: vertex angle α₂ = 2π/5 (base angle of triangle 1), base angles = (π - 2π/5)/2 = 3π/10 each.Triangle 3: vertex angle α₃ = 3π/10 (base angle of triangle 2), base angles = (π - 3π/10)/2 = 7π/20 each.Triangle 4: vertex angle α₄ = 7π/20 (base angle of triangle 3), base angles = (π - 7π/20)/2 = 13π/40 each.Wait, so α₄ is 7π/20, and the next base angles would be 13π/40. So, the pattern is that each αₙ is the base angle of the previous triangle, which is (π - αₙ₋₁)/2.So, in general, αₙ = (π - αₙ₋₁)/2.Wait, but in the first step, α₂ was 2π/5, which is the base angle of triangle 1, which is (π - α₁)/2 = (π - π/5)/2 = 2π/5. So, yes, α₂ = (π - α₁)/2.Similarly, α₃ = (π - α₂)/2 = (π - 2π/5)/2 = 3π/10.α₄ = (π - α₃)/2 = (π - 3π/10)/2 = 7π/20.So, the recurrence relation is αₙ = (π - αₙ₋₁)/2.Now, to find a general formula for αₙ, we can solve this recurrence relation.This is a linear recurrence relation of the form αₙ + (1/2)αₙ₋₁ = π/2.To solve this, we can find the homogeneous solution and a particular solution.First, the homogeneous equation is αₙ + (1/2)αₙ₋₁ = 0.The characteristic equation is r + 1/2 = 0, so r = -1/2.So, the homogeneous solution is αₙ^h = C(-1/2)^n.Now, for the particular solution, since the nonhomogeneous term is a constant (π/2), we can assume a constant particular solution αₙ^p = K.Substituting into the recurrence relation:K + (1/2)K = π/2(3/2)K = π/2K = (π/2) * (2/3) = π/3.So, the general solution is αₙ = αₙ^h + αₙ^p = C(-1/2)^n + π/3.Now, we can use the initial condition to find C. Let's use n=1: α₁ = π/5.So, π/5 = C(-1/2)^1 + π/3π/5 = -C/2 + π/3Let's solve for C:-C/2 = π/5 - π/3 = (3π - 5π)/15 = (-2π)/15So, C/2 = 2π/15C = 4π/15Therefore, the general formula is:αₙ = (4π/15)(-1/2)^n + π/3.We can write this as:αₙ = π/3 + (4π/15)(-1/2)^n.So, that's the general formula for αₙ.Let me check this with the values we have:For n=1:α₁ = π/3 + (4π/15)(-1/2)^1 = π/3 - 4π/30 = π/3 - 2π/15 = (5π - 2π)/15 = 3π/15 = π/5. Correct.For n=2:α₂ = π/3 + (4π/15)(-1/2)^2 = π/3 + (4π/15)(1/4) = π/3 + π/15 = (5π + π)/15 = 6π/15 = 2π/5. Correct.For n=3:α₃ = π/3 + (4π/15)(-1/2)^3 = π/3 + (4π/15)(-1/8) = π/3 - π/30 = (10π - π)/30 = 9π/30 = 3π/10. Correct.For n=4:α₄ = π/3 + (4π/15)(-1/2)^4 = π/3 + (4π/15)(1/16) = π/3 + π/60 = (20π + π)/60 = 21π/60 = 7π/20. Correct.So, the formula works for the first four terms. Therefore, the general formula is αₙ = π/3 + (4π/15)(-1/2)^n.I think that's it.