Question

If $\left(x_{1},\, y_{1}\right)$ and $\left(x_{2}, \,y_{2}\right)$ are the end points of a focal chord of the parabola $y^{2}=5 x$, then $4 x_{1} \,x_{2}+y_{1}\, y_{2}$ is equal to

• A. 25
• B. 5
• C. 0
• D. $\frac{5}{4}$

Answer 1

Answer: (C)

$\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ are the end points of a focal chord of the parabola $y^{2}=5 x$
Let $x_{1}=\frac{5}{4}\, t_{1}^{2}$ and $y_{1}=\frac{5}{2} \,t_{1}$
$\therefore x_{2}=\frac{5}{4}\left(\frac{1}{t_{1}^{2}}\right)$ and $y_{2}=\frac{5}{2}\left(\frac{-1}{t_{1}}\right)$
$\left[\because t_{1} \,t_{2}=-1\right.$ as $t_{1}$ and $t_{2}$ are the end points of a focal chord]
Now, $ x_{1} \,x_{2} =\frac{25}{16} $
and $ y_{1} \,y_{2} =-\frac{25}{4} $
$ \therefore 4 x_{1} \,x_{2}+y_{1} \,y_{2} =4\left(\frac{25}{16}\right)-\frac{25}{4} $
$\frac{25}{4}-\frac{25}{4} =0$