question_answer

Let P be the point of intersection of the common tangents to the parabola \[{{y}^{2}}=12x\] and the hyperbola \[8{{x}^{2}}{{y}^{2}}=8.\]If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio:                                                                [JEE Main Held on 12-4-2019 Morning]

A) 5:4      

B) 14:13

C) 2:1                              

D) 13:11

Correct Answer: A

Solution :

Equation of tangents \[{{y}^{2}}=12x\Rightarrow y=2x+\frac{3}{m}\] \[\frac{{{x}^{2}}}{1}-\frac{{{y}^{2}}}{8}=1\Rightarrow y=mx\pm \sqrt{{{m}^{2}}-8}\] Since they are common tangent \[\therefore \]\[\frac{3}{m}=\pm \sqrt{{{m}^{2}}-8}\]      \[\frac{{{x}^{2}}}{1}-\frac{{{y}^{2}}}{8}=1\] \[{{m}^{4}}-8{{m}^{2}}-9=0\]  \[e=3\] \[m=\pm 3\]                            \[ae=3\]