Let P be the point of intersection of the common tangents to the parabola y^{2}=12x and the hyperbola 8x^{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 : 


 

  • Option 1)

    14:13

  • Option 2)

    5:4

  • Option 3)

    13:11

  • Option 4)

    2:1

 

Answers (1)
V Vakul

Parabola y^{2}=12x

hyperbola  8x^{2}-y^{2}=8

\frac{x^{2}}{1}-\frac{y^{2}}{8}=1

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}\: \: \: \: \: \: \: \: \: \: \: \:                                              Eccentricity of hyperbola

\frac{9}{m^{2}}=m^{2}-8\: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: e=\frac{\sqrt{a^{2}+b^{2}}}{a}

m^{4}-8m^{2}-9=0\: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =\frac{\sqrt{1+8}}{1}

m^{4}-\left ( 9-1 \right )m^{2}-9=0\: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =3

\left ( m^{2}-9 \right )\left ( m^{2}+1 \right )=0\: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:

m=\pm 3\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: e=3

\therefore y=3x+1\: \: \: \: \: \: P\left ( -\frac{1}{3},0 \right )

y=-3x-1

S=\left ( 3,0 \right )

S'=\left (- 3,0 \right )

So ratio=\frac{10}{3}\times \frac{3}{8}=\frac{5}{4}

5:4

 


Option 1)

14:13

Option 2)

5:4

Option 3)

13:11

Option 4)

2:1

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