JEE Mathematics 2024 Question with Solution

Given: The circle of minimum area touches the parabola $y = 6 - x^2$ and the lines $y = \sqrt{3}|x|$.

Find: Which given point lies on the circle.

By symmetry, the center of the circle lies on the $y$-axis. If the radius is $r$, then the circle touches the parabola at its top side, so its center is $\left(0, 6-r\right)$.

Hence the equation of the circle is

$$x^2 + \left(y - (6-r)\right)^2 = r^2$$

The perpendicular distance from the center $\left(0, 6-r\right)$ to the line $y = \sqrt{3}x$ must be equal to the radius $r$.

So,

$$\frac{|0-(6-r)|}{\sqrt{1^2 + (\sqrt{3})^2}} = r$$

This gives

$$\frac{|6-r|}{2} = r$$

Now solve:

$$6-r = 2r \Rightarrow 6 = 3r \Rightarrow r = 2$$

The other case,

$$6-r = -2r \Rightarrow 6 = -r \Rightarrow r = -6$$

is not valid because radius is positive.

Therefore, $r = 2$ and the center is $\left(0, 4\right)$.

So the circle is

$$x^2 + (y-4)^2 = 4$$

Now check the options. For $\left(2,4\right)$,

$$2^2 + (4-4)^2 = 4$$ $$4 + 0 = 4$$

So $\left(2,4\right)$ lies on the circle.

Therefore, the correct option is A.

Given: The parabola $y = 6 - x^2$ and the pair of lines $y = \sqrt{3}|x|$ are symmetric about the $y$-axis.

Find: The point that lies on the smallest tangent circle.

Because the figure is symmetric, the required circle also has its center on the $y$-axis. Let its center be $\left(0, 6-r\right)$ and radius be $r$. The top contact with the parabola occurs at the highest point of the circle, matching the vertical placement used in the solution.

Tangency with each line means the distance from the center to either line equals $r$. Using $y = \sqrt{3}x$,

$$\frac{|6-r|}{2} = r$$

This gives the only positive solution $r = 2$.

Hence the circle becomes

$$x^2 + (y-4)^2 = 4$$

Testing the points:

• $(2,4): 4 + 0 = 4$, so it lies on the circle.
• $(1,2): 1 + 4 = 5$, so it does not lie on the circle.
• $(2,2): 4 + 4 = 8$, so it does not lie on the circle.
• $(1,1): 1 + 9 = 10$, so it does not lie on the circle.

Therefore, the point on the circle is $(2,4)$, so the correct option is A.