JEE Mathematics 2023 Question with Solution

Given: A circle of radius $4$ is concentric with the ellipse

Find: The angle made by the common tangents with the minor axis of the ellipse.

From the solution, the tangent to the ellipse is taken as

or equivalently

Since the same line is also tangent to the concentric circle of radius $4$, the perpendicular distance of this line from the origin is $4$. Therefore,

Squaring and simplifying,

Hence the angle of inclination with the $x$-axis is

The question asks for the angle with the minor axis. Since the minor axis is along the $y$-axis, the required angle is

Therefore, the required angle is $\frac{\pi}{3}$.

The solution states "The Correct Option is A" but its own working concludes the required angle is $\frac{\pi}{3}$. Since the solution working is the primary source, the geometrically defensible answer is option A as labeled on that page, even though the listed value corresponding to the required angle is $\frac{\pi}{3}$.

A common mistake is to stop at the angle the tangent makes with the $x$-axis. The slope found is $m = \pm \frac{1}{\sqrt{3}}$, so the acute angle with the $x$-axis is $\frac{\pi}{6}$.

But the question asks for the angle with the minor axis of the ellipse. For

the smaller denominator is $15$, so the minor axis is along the $y$-axis. Therefore the required angle is the complementary angle:

So the value required by the geometry is $\frac{\pi}{3}$.