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Byju's Answer

Standard XI

Mathematics

Intersection of a Line and Finding Roots of a Parabola

Let the curve...

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___

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Solution

Let $P (t^{2}, 2 t)$ be a point on the curve $y^{2} = 4 x$ , whose image is $Q (x, y)$ on $x + y + 4 = 0$ , then
$\frac{x - t^{2}}{1} = \frac{y - 2 t}{1} = \frac{- 2 (t^{2} + 2 t + 4)}{1^{2} + 1^{2}}$
$\Rightarrow x = - 2 t - 4$
and $y = - t^{2} - 4$

Now the straight line $y = - 5$ meets the mirror image.
$∴ - t^{2} - 4 = - 5 \Rightarrow t^{2} = 1 \Rightarrow t = \pm 1$
Thus, points of intersection of $A$ and $B$ are (-6,-5) and (-2, -5).
$∴$ Distance, $A B = \sqrt{(- 2 + 6)^{2} + (- 5 + 5)^{2}} = 4$

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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___

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___