JEE Main 2025 — Parabola Question with Solution

Given the parabola $y^2=16x$, the point $P$ on the focal chord $PQ$ is $(1,-4)$. We need to determine how the focus divides the chord.

First, express the coordinates for the point $P$ using the parametric form of the parabola. The general parametric equations for the parabola are:

$P(a{t}^2,2at)\equiv P(4t^2,8t)\equiv (1,-4)$

The equation $8t=-4$ leads to solving for $t$:

$t=-\frac{1}{2}$

Now, find the coordinates of point $Q$ using the relation $t_1\cdot t_2=-1$. Therefore:

$Q\left(\frac{a}{t^2},\frac{-2a}{t}\right)$

Given the focus $S$ of the parabola is at $(4,0)$, we can find the lengths of $PS$ and $QS$:

The length $PS$ is calculated as:

$PS=a+a{t}^2$

And $QS$ is given by:

$QS=a+\frac{a}{t^2}=\frac{a{t}^2+a}{t^2}$

Since the focus divides the chord in a specific ratio $m:n$, and $\gcd (m,n)=1$, we find:

$\frac{PS}{QS}=t^2=\frac{1}{4}=\frac{m}{n}$

Thus, the squares of $m$ and $n$ sum up to:

$m^2+n^2=17$