Question

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

• A. $\pm 1$
• B. $\frac{4}{3}$
• C. $\frac{3}{4}$
• D. None of these

Answer 1

The correct option is A $\pm 1$

Let the focus be $(a,b)$.
Equation of parabola with focus $(a,b)$ and directrix is y-axis
$S_1:(x-a{)}^2+(y-b{)}^2=x^2$
Equation of parabola with focus $(a,b)$ and directrix is x-axis.
$S_2:(x-a{)}^2+(y-b{)}^2=y^2$
Equation of common chord is
$S_1-S_2=0$ gives
$x^2-y^2=0$
$y=\pm x$