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 $y^2=5x$, then $4x_1{x}_2+y_1{y}_2$ 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 $y^2=5x$
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 $y_2=\frac{5}{2}\left(\frac{-1}{t_1}\right)$
$[\because t_1{t}_2=-1$ as $t_1$ and $t_2$ are the end points of a focal chord]
Now, $x_1{x}_2=\frac{25}{16}$
and $y_1{y}_2=-\frac{25}{4}$
$\therefore 4x_1{x}_2+y_1{y}_2=4\left(\frac{25}{16}\right)-\frac{25}{4}$
$\frac{25}{4}-\frac{25}{4}=0$