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

Question

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

Accepted Answer

Let intersection points of these two parabolas are $$\mathrm{A}\left(\mathrm{x}_1, \mathrm{y}_1\right) \& \mathrm{~B}\left(\mathrm{x}_2, \mathrm{y}_2\right)$$

$\because$ equation of parabola I and II are given below

$$\begin{aligned}
& \therefore(\mathrm{x}-4)^2+(\mathrm{y}-3)^2=\mathrm{x}^2 \quad\text{..... (1)}\
& &(\mathrm{x}-4)^2+(\mathrm{y}-3)^2=\mathrm{y}^2 \quad\text{..... (2)}
\end{aligned}$$

Here $A\left(x_1, y_1\right) \& B\left(x_2, y_2\right)$ will satisfy the equation

Also from equations (1) \& (2), we get $x=y$ .......(3)

Put $x=y$ in equation (1)

We get $x^2-14 x+25=0$

$$\begin{aligned}
& x_1+x_2=14 \
& x_1 x_2=25 \
& \therefore A B^2=\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2 \
& =2\left(x_1-x_2\right)^2 \
& =2\left[\left(x_1+x_2\right)^2-4 x_1 x_2\right] \
& =192
\end{aligned}$$

Solutions · 1