JEE Mathematics 2025 Question with Solution

Given: The parabola is $y^2 = 4x$. The focal chord $PQ$ makes an angle of $60^\circ$ with the positive x-axis, and $P$ lies in the first quadrant. A circle with diameter $PS$, where $S$ is the focus, touches the y-axis at $(0, \alpha)$.

Find: The value of $5\alpha^2$.

For $y^2 = 4ax$, we have $a = 1$, so the focus is $S(1,0)$.

Take the parametric point

$$P(t^2, 2t)$$

on the parabola.

Since the focal chord passes through $S$ and makes angle $60^\circ$ with the positive x-axis, its slope is

$$\tan 60^\circ = \sqrt{3}$$

The slope of $PS$ is

$$\frac{2t - 0}{t^2 - 1} = \frac{2t}{t^2 - 1}$$

So,

$$\frac{2t}{t^2 - 1} = \sqrt{3}$$

Hence,

$$2t = \sqrt{3}(t^2 - 1)$$ $$\sqrt{3}t^2 - 2t - \sqrt{3} = 0$$

Solving,

$$t = \frac{2 \pm 4}{2\sqrt{3}}$$

Thus,

$$t = \sqrt{3} \quad \text{or} \quad t = -\frac{1}{\sqrt{3}}$$

Since $P$ lies in the first quadrant, $t > 0$, so

$$t = \sqrt{3}$$

Therefore,

$$P = (3, 2\sqrt{3})$$

Now the circle has diameter joining $P(3,2\sqrt{3})$ and $S(1,0)$. Its center is the midpoint:

$$\left(\frac{3+1}{2}, \frac{2\sqrt{3}+0}{2}\right) = (2, \sqrt{3})$$

Its radius is half of $PS$, or equivalently the distance from the center to $S$:

$$r = \sqrt{(2-1)^2 + (\sqrt{3}-0)^2}$$ $$r = \sqrt{1+3} = 2$$

Since the circle touches the y-axis, the point of contact has y-coordinate equal to the y-coordinate of the center. Hence,

$$\alpha = \sqrt{3}$$

Therefore,

$$5\alpha^2 = 5(\sqrt{3})^2 = 15$$

So, the correct option is A.

Given: $P(t^2,2t)$ on the parabola $y^2=4x$ and focus $S(1,0)$.

Find: $5\alpha^2$.

The slope of the focal chord is $\sqrt{3}$, so

$$\frac{2t}{t^2-1} = \sqrt{3}$$

This gives the positive value

$$t = \sqrt{3}$$

Hence,

$$P = (3,2\sqrt{3})$$

The center of the circle with diameter $PS$ is the midpoint:

$$C = (2,\sqrt{3})$$

A circle touching the y-axis at $(0,\alpha)$ has point of tangency vertically aligned with its center, so

$$\alpha = \sqrt{3}$$

Thus,

$$5\alpha^2 = 5 \cdot 3 = 15$$

Therefore, the correct option is A.