Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersects at the points $A$ and $B$, then $(A B)^2$ is equal to :

Question

Aicharya · Accepted Answer

The parabolas are
$(x-4)^2+(y-3)^2=x^2$ ...(i)
and $(x-4)^2+(y-3)^2=y^2$...
If point of intersection are $A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ By solving (i) and (ii), we get
$\begin{aligned}
& x_1+x_2=14 \text { and } x_1 x_2=25 \\ & (A B)^2=2\left(\left(x_1+x_2\right)^2-4 x_1 x_2\right)=192
\end{aligned}$

Two parabolas have the same focus \((4,3)\) and their directrices are the \(x\)-axis and the \(y\)-axis,…

Solution