Let the curve be the mirror image of the parabola with respect to the line If A and B are the points of intersection of with the line , then the distance between A and B is [2015]

Question

Let the curve $C$ be the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0 .$ If A and B are the points of intersection of $C$ with the line $y = - 5$ , then the distance between A and B is [2015]

Smart Achievers · Accepted Answer

(4)

Let $y^{2} = 4 x$ be any point on $y^{2} = 4 x$ and $(h, k)$ be the image of $(t^{2}, 2 t)$ in the line $x + y + 4 = 0 .$

$∴$ $\frac{h - t^{2}}{1} = \frac{k - 2 t}{1} = \frac{- 2 (t^{2} + 2 t + 4)}{2}$

$\Rightarrow h = - (2 t + 4)$ and $k = - (t^{2} + 4)$

For its intersection with $y = - 5$ , we get

$- (t^{2} + 4) = - 5 \Rightarrow t = \pm 1$

$∴$ $A (- 6, - 5) and B (- 2, - 5),$ $∴$ $A B = 4 .$

Solution

Q.
Let the curve $C$ be the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0 .$ If A and B are the points of intersection of $C$ with the line $y = - 5$ , then the distance between A and B is [2015]

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Solution

Q. Let the curve C be the mirror image of the parabola y2=4x with respect to the line x+y+4=0. If A and B are the points of intersection of C with the line y=-5, then the distance between A and B is [2015]

Ans. (4) Let (t2,2t) be any point on y2=4x and (h,k) be the image of (t2,2t) in the line x+y+4=0. ∴ h-t21=k-2t1=-2(t2+2t+4)2 ⇒h=-(2t+4) and k=-(t2+4) For its intersection with y=-5, we get -(t2+4)=-5⇒t=±1 ∴ A(-6,-5) and B(-2,-5), ∴ AB=4.

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Q.
Let the curve $C$ be the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0 .$ If A and B are the points of intersection of $C$ with the line $y = - 5$ , then the distance between A and B is [2015]