question_answer

The locus of the point of intersection of two tangents to the parabola \[{{y}^{2}}=4ax,\] which are at right angle to one another is

A) \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]

B) \[a{{y}^{2}}=x\]

C) \[x+a=0\]

D) \[x+y\pm a=0\]

Correct Answer: C

Solution :

[c] Let the two tangents to the parabola \[{{y}^{2}}=4ax\]be PT and QT which are at right angle to one another at \[T(h,k)\]. Then we have to find the locus of T (h, k).

We know that \[y=mx+\frac{a}{m},\] where m is the slope is the equation of tangent to the parabola \[{{y}^{2}}=4ax\]for all m.

Since this tangent to the parabola will pass through \[T(h,k)\] so

\[k=mh+\frac{a}{m};\] Or \[{{m}^{2}}h-mk+a=0\]

This is a quadratic equation in m so will have

Two roots, say \[{{m}_{1}}\,\,and\,\,{{m}_{2}},\] then

\[{{m}_{1}}+{{m}_{2}}=\frac{k}{h},\] and \[{{m}_{1}}:{{m}_{2}}=\frac{a}{h}\]

Given that the two tangents intersect at right angle so \[{{m}_{1}},{{m}_{2}}=-1\] or \[\frac{a}{h}=-1\] or \[h+a=0\]

The locus of \[T(h,k)\] is \[x+a=0,\] which is the equation of directrix.