JEE Mathematics 2023 Question with Solution

Given: The ellipse is

$$\frac{x^2}{36} + \frac{y^2}{4} = 1$$

and the point is $\left(3\sqrt{3}, 1\right)$.

Find: The value of $\alpha^2 - \beta^2$ where $(\alpha, \beta)$ is the intersection point of tangents at P and Q on the circle with diameter AB.

The tangent to

$$\frac{x^2}{36} + \frac{y^2}{4} = 1$$

at $\left(x_1, y_1\right)$ is

$$\frac{x x_1}{36} + \frac{y y_1}{4} = 1$$

So at $\left(3\sqrt{3}, 1\right)$,

$$\frac{x \cdot 3\sqrt{3}}{36} + \frac{y}{4} = 1$$

which becomes

$$\frac{x\sqrt{3}}{12} + \frac{y}{4} = 1$$

Putting $x=0$, we get $y=4$. Therefore,

$$A = (0,4)$$

The slope of the tangent is obtained by rewriting it as

$$y = 4 - \frac{\sqrt{3}}{3}x$$

Hence the slope of the tangent is

$$-\frac{\sqrt{3}}{3}$$

Therefore, the slope of the normal is

$$\sqrt{3}$$

So the normal through $\left(3\sqrt{3},1\right)$ is

$$y - 1 = \sqrt{3}\left(x - 3\sqrt{3}\right)$$

Putting $x=0$,

$$y - 1 = -9$$

so

$$y = -8$$

Thus,

$$B = (0,-8)$$

Now the circle with diameter AB has center at the midpoint of A and B:

$$O = \left(0,-2\right)$$

Its radius is

$$6$$

Hence its equation is

$$x^2 + (y+2)^2 = 36$$

The line $x = 2\sqrt{5}$ cuts the circle at P and Q. Substituting in the circle,

$$(2\sqrt{5})^2 + (y+2)^2 = 36$$ $$20 + (y+2)^2 = 36$$ $$(y+2)^2 = 16$$

So

$$y+2 = \pm 4$$

Hence,

$$y=2 \quad \text{or} \quad y=-6$$

Therefore,

$$P = (2\sqrt{5},2), \quad Q = (2\sqrt{5},-6)$$

For the circle

$$x^2 + (y+2)^2 = 36$$

the tangent at $\left(x_1, y_1\right)$ is

$$xx_1 + (y+2)(y_1+2) = 36$$

At $P=(2\sqrt{5},2)$,

$$2\sqrt{5}x + 4(y+2) = 36$$

At $Q=(2\sqrt{5},-6)$,

$$2\sqrt{5}x - 4(y+2) = 36$$

Adding the two equations,

$$4\sqrt{5}x = 72$$ $$x = \frac{18}{\sqrt{5}} = \frac{18\sqrt{5}}{5}$$

Substituting back,

$$36 + 4(y+2) = 36$$

so

$$y = -2$$

Thus,

$$(\alpha, \beta) = \left(\frac{18\sqrt{5}}{5}, -2\right)$$

Now compute

$$\alpha^2 = \left(\frac{18\sqrt{5}}{5}\right)^2 = \frac{324}{5}, \qquad \beta^2 = (-2)^2 = 4$$

Therefore,

$$\alpha^2 - \beta^2 = \frac{324}{5} - 4 = \frac{304}{5}$$

So the correct option is A.

Note: The solution marks option C, but the worked steps clearly give $\frac{304}{5}$, which matches option A.