💡First, I need to find the total sum of all the numbers provided: (36, 27, 42, 32, 28, 31, 23, 17).Adding them together:[36 + 27 + 42 + 32 + 28 + 31 + 23 + 17 = 236]Since there are 8 numbers, they will form 4 pairs. To find the sum of each pair, I divide the total sum by 4:[frac{236}{4} = 59]Each pair must sum to 59. To find the number paired with 32, I subtract 32 from 59:[59 - 32 = 27]Therefore, the number paired with 32 is 27.

💡Okay, so I have this problem here about triangles, and I need to figure out which one isn't a right triangle. The options are A, B, C, and D. Let me go through each one step by step.Starting with option A: The ratio of the three interior angles is 1:2:3. Hmm, I remember that in a triangle, the sum of the interior angles is always 180 degrees. So if the angles are in the ratio 1:2:3, I can let the measures be x, 2x, and 3x. Adding them up: x + 2x + 3x = 6x. So 6x = 180 degrees, which means x = 30 degrees. Therefore, the angles are 30, 60, and 90 degrees. Oh, wait, that means one of the angles is 90 degrees, so it is a right triangle. So option A is a right triangle, so it's not the answer we're looking for.Moving on to option B: The lengths of the three sides are 1, sqrt(3), and 2. I think this is a classic Pythagorean triple. Let me check: 1 squared is 1, sqrt(3) squared is 3, and 2 squared is 4. So 1 + 3 equals 4, which is equal to 2 squared. That means this triangle satisfies the Pythagorean theorem, so it's a right triangle. So option B is also a right triangle, not the answer.Option C: The ratio of the lengths of the three sides is 3:4:5. I remember that 3:4:5 is a well-known Pythagorean triple. Let me verify: 3 squared is 9, 4 squared is 16, and 5 squared is 25. So 9 + 16 equals 25, which is equal to 5 squared. Yep, that's a right triangle as well. So option C is a right triangle, so it's not the answer.Finally, option D: The ratio of the three interior angles is 3:4:5. Again, the sum of the interior angles is 180 degrees. Let me denote the measures as 3x, 4x, and 5x. Adding them up: 3x + 4x + 5x = 12x. So 12x = 180 degrees, which means x = 15 degrees. Therefore, the angles are 45, 60, and 75 degrees. None of these angles is 90 degrees, so this triangle isn't a right triangle. Wait, let me double-check that. If the angles are 45, 60, and 75, none is 90, so yeah, it's not a right triangle. So option D is the one that isn't a right triangle.I think that's it. So the answer should be D.**Final Answer**boxed{D}

💡To determine the value of ( x ) such that the points ( A(-1, 1) ), ( B(2, -4) ), and ( C(x, -9) ) are collinear, we can use the concept of slope. Since the points are collinear, the slope between any two pairs of points should be the same.First, calculate the slope between points ( A ) and ( B ):[text{Slope of } AB = frac{y_B - y_A}{x_B - x_A} = frac{-4 - 1}{2 - (-1)} = frac{-5}{3}]Next, calculate the slope between points ( B ) and ( C ):[text{Slope of } BC = frac{y_C - y_B}{x_C - x_B} = frac{-9 - (-4)}{x - 2} = frac{-5}{x - 2}]Since the slopes must be equal for the points to be collinear:[frac{-5}{3} = frac{-5}{x - 2}]Solving for ( x ):[frac{-5}{3} = frac{-5}{x - 2} Rightarrow frac{1}{3} = frac{1}{x - 2} Rightarrow x - 2 = 3 Rightarrow x = 5]

💡First, I need to simplify the left side of the equation by combining like terms. Since all the terms are the same (2²ⁿ), I can add them together by multiplying the term by the number of times it appears, which is 4 times in this case.So, the left side of the equation becomes:4 * (2²ⁿ)Next, I substitute n = 17 into the equation:4 * (2²¹⁷)This simplifies to:4 * (2³⁴)Now, the equation looks like this:4 * (2³⁴) = some number¹⁸To find the base number on the right side of the equation, I need to express the left side as a power of a single base to the 18th power. Since 4 is 2 squared, I can rewrite 4 as 2²:(2²) * (2³⁴)Using the property of exponents that states aᵐ * aⁿ = aᵐ⁺ⁿ, I can add the exponents:2²⁺³⁴ = 2³⁶Now I have:2³⁶ = some number¹⁸To express the left side as a power of 18, I need to find a power of 2 that equals 18. Since 36 is twice 18, I can take the square root of 2³⁶ to get the base number to the 18th power:(2³⁶)^(1/2) = (some number)¹⁸This simplifies to:2¹⁸ = some number¹⁸Therefore, the base number on the right side of the equation is 2.

💡Alright, so I have this problem here about a parabola with a given directrix, and I need to find its standard equation from the given options. Let me try to figure this out step by step.First, I remember that a parabola is the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. So, if I know the directrix, I can find the focus and then determine the equation of the parabola.The directrix given is ( x = frac{1}{2} ). Hmm, that's a vertical line, right? So, if the directrix is vertical, the parabola must open either to the left or to the right. I think when the directrix is vertical, the parabola opens horizontally. Now, I need to figure out whether it opens to the left or the right.I recall that for a parabola that opens to the right, the directrix is on the left side of the vertex, and for one that opens to the left, the directrix is on the right side of the vertex. Since the directrix here is ( x = frac{1}{2} ), which is to the right of the origin, I think the parabola must open to the left. That makes sense because the focus would then be on the opposite side, to the left of the vertex.Now, let's recall the standard form of a parabola that opens to the left or right. I think it's ( y^2 = 4px ), where ( p ) is the distance from the vertex to the focus (and also from the vertex to the directrix). If the parabola opens to the right, ( p ) is positive, and if it opens to the left, ( p ) is negative.Given that the directrix is ( x = frac{1}{2} ), the vertex of the parabola should be halfway between the focus and the directrix. Since the directrix is at ( x = frac{1}{2} ), and assuming the vertex is at the origin (0,0) because the options given are all in terms of ( x ) and ( y ) without any shifts), the distance from the vertex to the directrix is ( frac{1}{2} ). Therefore, ( p = -frac{1}{2} ) because it's opening to the left.Wait, let me make sure. If the directrix is ( x = frac{1}{2} ), and the vertex is at (0,0), then the distance from the vertex to the directrix is indeed ( frac{1}{2} ). Since the parabola opens to the left, ( p ) should be negative, so ( p = -frac{1}{2} ).Now, plugging ( p ) into the standard equation ( y^2 = 4px ), we get:( y^2 = 4 times (-frac{1}{2}) times x )Simplifying that:( y^2 = -2x )Looking at the options, option D is ( y^2 = -2x ), which matches what I got. So, that should be the correct answer.But wait, let me double-check. Sometimes I get confused between the directrix and the focus. Let me recall that for a parabola opening to the left, the focus is at ( (p, 0) ), which in this case would be ( (-frac{1}{2}, 0) ). The directrix is ( x = -p ), which would be ( x = frac{1}{2} ). Yes, that matches the given directrix. So, everything checks out.I think I'm confident now that the correct equation is ( y^2 = -2x ), which is option D.

💡Alright, so I've got this problem about an ellipse and a line. It's divided into two parts: first, finding the ordinary equations of the line and the ellipse, and second, finding the sum of the distances from the foci of the ellipse to the line. Hmm, okay, let me take it step by step.Starting with part (I). The problem gives me a polar equation for the ellipse: ρ² = 12 / (3cos²θ + 4sin²θ). I need to convert this into the standard Cartesian form. I remember that in polar coordinates, ρ² = x² + y², and cosθ = x/ρ, sinθ = y/ρ. So maybe I can substitute these into the equation.Let me write that down:ρ² = 12 / (3cos²θ + 4sin²θ)Substituting cosθ and sinθ:ρ² = 12 / [3(x/ρ)² + 4(y/ρ)²]Simplify the denominator:3(x²/ρ²) + 4(y²/ρ²) = (3x² + 4y²)/ρ²So now the equation becomes:ρ² = 12 / [(3x² + 4y²)/ρ²] = 12ρ² / (3x² + 4y²)Wait, so ρ² = 12ρ² / (3x² + 4y²). Let me rearrange this:Multiply both sides by (3x² + 4y²):ρ²(3x² + 4y²) = 12ρ²Divide both sides by ρ² (assuming ρ ≠ 0, which it isn't for an ellipse):3x² + 4y² = 12Hmm, that looks like an ellipse equation. To write it in standard form, I should divide both sides by 12:(3x²)/12 + (4y²)/12 = 1Simplify:x²/4 + y²/3 = 1Okay, so that's the standard equation of the ellipse. Good, so part (I) for the curve C is done.Now, the parametric equation of line l is given as:x = 2 + (√2/2)ty = (√2/2)tI need to convert this into the ordinary (Cartesian) equation. For parametric equations, I can solve for t in one equation and substitute into the other.From the y equation:y = (√2/2)t => t = (2/√2)y = √2 yNow plug this into the x equation:x = 2 + (√2/2)(√2 y) = 2 + (2/2)y = 2 + ySo, x = y + 2 => y = x - 2That's the equation of the line. So, part (I) is done: the line is y = x - 2 and the ellipse is x²/4 + y²/3 = 1.Moving on to part (II). I need to find the sum of the distances from the foci F₁ and F₂ to the line l.First, I should find the coordinates of the foci of the ellipse. The standard form of the ellipse is x²/a² + y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis. In this case, a² = 4 and b² = 3, so a = 2 and b = √3.For an ellipse, the distance from the center to each focus is c, where c² = a² - b². Let me compute that:c² = 4 - 3 = 1 => c = 1Since the ellipse is centered at the origin (because the equation is x²/4 + y²/3 = 1), the foci are located at (±c, 0) = (±1, 0). So, F₁ is (-1, 0) and F₂ is (1, 0).Now, I need to find the distance from each focus to the line l: y = x - 2.The formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / √(A² + B²). So, first, let me write the line l in standard form.Given y = x - 2, subtract x and add 2 to both sides:-x + y + 2 = 0So, A = -1, B = 1, C = 2.Now, compute the distance from F₁(-1, 0):d₁ = |A*(-1) + B*0 + C| / √(A² + B²) = |-1*(-1) + 1*0 + 2| / √(1 + 1) = |1 + 0 + 2| / √2 = |3| / √2 = 3/√2Similarly, compute the distance from F₂(1, 0):d₂ = |A*1 + B*0 + C| / √(A² + B²) = |-1*1 + 1*0 + 2| / √2 = |-1 + 0 + 2| / √2 = |1| / √2 = 1/√2So, the sum of the distances is d₁ + d₂ = 3/√2 + 1/√2 = 4/√2Simplify 4/√2: multiply numerator and denominator by √2 to rationalize:4√2 / 2 = 2√2So, the sum of the distances is 2√2.Wait, let me double-check my steps to make sure I didn't make a mistake.First, converting the polar equation: yes, I substituted correctly and ended up with x²/4 + y²/3 = 1. That seems right.For the parametric equations, solving for t in y and substituting into x gives x = y + 2, which simplifies to y = x - 2. That looks correct.Finding the foci: since a² = 4, b² = 3, c² = 1, so c = 1. So foci at (-1, 0) and (1, 0). Correct.Distance formula: line is -x + y + 2 = 0, so A = -1, B = 1, C = 2.For F₁(-1, 0):d₁ = |-1*(-1) + 1*0 + 2| / √(1 + 1) = |1 + 0 + 2| / √2 = 3/√2For F₂(1, 0):d₂ = |-1*1 + 1*0 + 2| / √2 = |-1 + 0 + 2| / √2 = 1/√2Sum: 3/√2 + 1/√2 = 4/√2 = 2√2. Yep, that seems correct.I think I did everything right. So the sum is 2√2.**Final Answer**The sum of the distances is boxed{2sqrt{2}}.