Question

If $(x_1,y_1)$ and $(x_2,y_2)$ are the extremeties of the focal chord of parabola $y^2=16ax$, then the value of $16x_1{x}_2+y_1{y}_2$ is

Answer 1

Let $(x_1,y_1)=(a{t}_1^2,4a{t}_1)$ and $(x_2,y_2)=(a{t}_2^2,4a{t}_2)$
For extremities of focal chord of the parabola we know that $t_1{t}_2=-1$
$16x_1{x}_2+y_1{y}_2=16a^2{t}_1^2{t}_2^2+16a^2{t}_1{t}_2=16-16=0$