Question

Assertion :The equation of common tangent to the parabola's $y^2=4x$ & $x^2=-32y$ is $x-2y+4=0$. Reason: If any line touches a curve it will cut the curve in two coincident points.

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Any tangent to $y^2=4ax$ is $y=mx+\frac{1}{m}$ as in $y^2=4\left(1\right)x(a=1)$.

If it touches $x^2=-32y$ then $x^2=-32(mx+\frac{1}{m})i.e.x^2+32mx+\frac{32}{m}=0$
will have equal roots

$\therefore D=(32m{)}^2-4\left(\frac{32}{m}\right)=0$

$\Rightarrow 8m^3-1=0\Rightarrow m=\frac{1}{2}$

$\therefore$ Required equation of common tangent is

$y=\frac{1}{2}x+2$ or $x-2y+4=0$

So, Assertion is true followed by Reason (R),

Hence, choice (a) is correct answer.