First slide

Parabola

Question

Let $L_{1}$ be the length of the common chord of the curves $x^{2} + y^{2} = 9$ and $y^{2} = 8 x$ , and $L_{2}$ be the length of the latus rectum of $y^{2} = 8 x$ , then

Moderate

A

$L_{1} > L_{2}$
B

$L_{1} = L_{2}$
C

$L_{1} < L_{2}$
D

$\frac{L_{1}}{L_{2}} = \sqrt{2}$

Solution

For the points of intersection of the curves
$x^{2} + y^{2} = 9$ and $y^{2} = 8 x$ , we have
$x^{2} + 8 x - 9 = 0 \Rightarrow x = - 9$ or $x = 1$
Since $x > 0$ for $y^{2} = 8 x$ , the points of intersection are $(1, 2 \sqrt{2})$
and $(1, - 2 \sqrt{2})$ so that the length of the common chord $L_{1}$ is $4 \sqrt{2}$ .
$L_{2} =$ length of the latus rectum of $y^{2} = 8 x$ is $8$ . So $L_{1} < L_{2}$

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