question_answer

Tangent and normal are drawn at \[p(16,16)\]on the parabola\[{{y}^{2}}=16x\], which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and \[\angle CPB=\theta \], then a value of tan \[\theta \]is:                        [JEE Main Online 08-04-2018]

A)  3                                

B)  \[\frac{4}{3}\]

C)  \[\frac{1}{2}\]                                   

D)  2

Correct Answer: D

Solution :

                \[A(-16,0)\] \[B(16+2(4),0)=B(24,0)\] \[AC=CB\Rightarrow C(4,0)\] C is the cimcumcentre \[BP=\sqrt{{{8}^{2}}+{{16}^{2}}}=8\sqrt{5}\] \[PC=\sqrt{{{12}^{2}}+{{16}^{2}}}=20\] \[PC=\sqrt{{{12}^{2}}+{{16}^{2}}}=20\] \[\Rightarrow \tan \theta =2\]