Question

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

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$\because y=x+\sqrt{5}$ is a common tangent.

so, statement I is true.

Now, let $y=mn+\frac{\sqrt{5}}{m}$ be a common tangent.

$\Rightarrow 2x^2+2y^2=5$

As tangent, $\left|\frac{m\times 0÷0+\frac{\sqrt{5}}{m}}{\sqrt{1+m^2}}\right|=\frac{\sqrt{5}}{2}$

$\Rightarrow m^2+m^2-2=0\Rightarrow m^2=1\Rightarrow m=\ne 1$

So, statement II is true.

But statement II is not a correct

explanation of statement I

So option (B) is correct.