First slide

Parabola

Question

Two parabolas have the same focus. If their directrices are the x and the y-axis, respectively, then the slope of their common chord is

Moderate

A

$\pm 1$
B

$\frac{4}{3}$
C

$\frac{3}{4}$
D

None of these

Solution

Let the focus be (a, b). The equations are
$\begin{array}{l} and \begin{aligned} S_{1} : (x - a)^{2} + (y - b)^{2} = x^{2} \\ S_{2} : (x - a)^{2} + (y - b)^{2} = y^{2} \end{aligned} \\ Common chord S_{1} - S_{2} = 0 gives \end{array}$
$x^{2} - y^{2} = 0$
$or y = \pm x$

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