Search any question & find its solution
Question:
Answered & Verified by Expert
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
Options:
Solution:
1095 Upvotes
Verified Answer
The correct answer is:
$y_1 y_2=a^2$
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.
$$
\begin{array}{rlrl}
& \therefore & t_1 t_2 & =-1 \\
\therefore & & y_1 y_2 & =\left(a t_1^2\right)\left(a t_2\right)^2 \\
& & =a^2\left(t_1 t_2\right)^2=a^2(-1)^2 \\
& & y_1 y_2 & =a^2
\end{array}
$$
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.
$$
\begin{array}{rlrl}
& \therefore & t_1 t_2 & =-1 \\
\therefore & & y_1 y_2 & =\left(a t_1^2\right)\left(a t_2\right)^2 \\
& & =a^2\left(t_1 t_2\right)^2=a^2(-1)^2 \\
& & y_1 y_2 & =a^2
\end{array}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.