Q57 / 90 · JEE Main 2021 (26 Feb Shift 2) · Mathematics

Let $L$ be a common tangent line to the curves $4 x^{2} + 9 y^{2} = 36$ and ${(2 x)}^{2} + {(2 y)}^{2} = 31$ . Then the square of the slope of the line $L$ is ______.

Correct answer: 3

Solution

Given curves are $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$

$x^{2} + y^{2} = \frac{31}{4}$

let slope of common tangent be $m$

so tangents are $y = m x \pm \sqrt{9 m^{2} + 4}$

$y = m x \pm \frac{\sqrt{31}}{2} \sqrt{1 + m^{2}}$

hence $9 m^{2} + 4 = \frac{31}{4} (1 + m^{2})$

$\Rightarrow 36 m^{2} + 16 = 31 + 31 m^{2} \Rightarrow m^{2} = 3$

Hence, the correct answer is 3.