If the coordinates of the ends of a focal chord of the parabola $x^2=4$ ay are $\left(x_1, y_1\right)$ and…
If the coordinates of the ends of a focal chord of the parabola $x^2=4$ ay are $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$, then
$y_1 y_2=4 a^2$
$y_1 y_2=-4 a^2$
$y_1 y_2=-a^2$
$y_1 y_2=a^2$
Solution
Given parabola, $x^2=4 a y$
Let point on parabola is $\left(2 a t_1, a t_1^2\right)$ and $\left(2 a t_2, a t_2^2\right)$
Now, $\left(2 a t_1, a t_1^2\right)$ and $\left(2 a t_2, a t_2^2\right)$ are end points of a focal chord of parabola.
$
\begin{array}{rlrl}
& \therefore & t_1 t_2 & =-1 \\
\therefore & & y_1 y_2 & =\left(a t_1^2\right)\left(a t_2\right)^2 \\
& & =a^2\left(t_1 t_2\right)^2=a^2(-1)^2 \\
& & y_1 y_2 & =a^2
\end{array}
$