question_answer

If tangents are drawn from any point on the line \[x+4a=0\] to the parabola \[{{y}^{2}}=4ax,\] then their chord of contact subtends angle at the vertex equal to

A) \[\frac{\pi }{4}\]

B) \[\frac{\pi }{3}\]

C) \[\frac{\pi }{2}\]

D) \[\frac{\pi }{6}\]

Correct Answer: C

Solution :

[c] Let \[R(-4a,k)\] be any point on the line \[x=-4a.\] The equation of chord of contact PQ w.r.t. \[P(-4a,k)\] is \[y.k=2a(x-4a)\]                                     ?. (1) Making equation of parabola \[{{y}^{2}}=4ax\] Homogeneous using (1), we get \[{{y}^{2}}=4ax\left( \frac{2ax-yk}{8{{a}^{2}}} \right)\] \[\Rightarrow 8{{a}^{2}}{{x}^{2}}-8{{a}^{2}}{{y}^{2}}-4akxy=0\] This represents the pair of straight lines AP and AQ. Since coefficient of \[{{x}^{2}}+\] coefficient of \[{{y}^{2}}=0\therefore \angle PAQ=90{}^\circ \] i.e., chord of contact PQ subtends a right angle at the vertex.