Question

Assertion :The tangents at the extremities of a focal cord of the parabola $y^2=4ax$ intersect on the line $x+a=0$. Reason: The locus of the point of intersection of perpendicular tangents to the parabola is its directrix.

• 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

Reason is true, perpendicular tangents to $y^2=4ax$ are $y=mx+\frac{a}{m},y=m^{\prime }x+\frac{a}{m^{\prime }}=-1$
They intersect at a point where $(m-m^{\prime })x=\frac{a(m-m^{\prime })}{m{m}^{\prime }}\Rightarrow x=-a$
i.e. the directrix of the parabola
Slope of the tangent at the extremities $t$ and $t^{\prime }$ of the parabola $y^2=4ax$ are $\frac{1}{t}$ and $\frac{1}{t^{\prime }}$
Product of the slopes is $\frac{1}{t{t}^{\prime }}=-1$
So, tangents are perpendicular and hence using reason assertion is true.