Question

Given: A circle, $2x^2+2y^2=5$ and a parabola, $y^2=4\sqrt{5}x$.
$\text{Statement-I}:$ An equation of a common tangent to these curves is $y=x+\sqrt{5}$
$\text{Statement-II}:$ If the line, $y=mx+\frac{\sqrt{5}}{m}(m\ne 0)$ is their common tangent, then $m$ satisfies $m^4-3m^2+2=0$.

• A. Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
• B. Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
• C. Statement-I is true; Statement-II is false.
• D. Statement-I is false; Statement-II is true.

Answer 1

The correct option is B Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.


The tangent on the parabola $y^2=4\sqrt{5}x$ is
$y=mx+\frac{\sqrt{5}}{m}...\left(1\right)$
which is also tangent on circle

$OP=r$
$\Rightarrow \frac{\frac{\sqrt{5}}{m}}{\sqrt{1+m^2}}=\sqrt{\frac{5}{2}}$
$\Rightarrow m^4+m^2-2=0$
$\Rightarrow (m^2+2)(m^2-1)=0$
$\Rightarrow m=\pm 1$
Thus, from equation $\left(1\right)$
$y=\pm (x+\sqrt{5})$
$m=\pm 1$ is also satisfying $m^4-3m^2+2=0$
So, both Statement-I and Statement-II are true but Statement-II is not a correct explanation for Statement-I