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Let P be the point of intersection of the common tangents to the parabola y2=12x and the hyperbola 8x2y2=8. If S and S denote the foci of the hyperbola where S lies on the positive xaxis then P divides SS in a ratio
  1. 14:13
  2. 13:11
  3. 5:4
  4. 2:1


Solution

The correct option is C 5:4
Equation of parabola y2=12x
So equation of its tangent : y=3x+3m
Equation of hyperbola x21y28=1
Eccentricity of  hyperbola e=1+8=3
S(ae,0)=S(3,0)   &   S(ae,0)=S(3,0)

And equation of its tangent : y=mx±m28
Both tangent are coomon tangents
Therefore, 9m2=m28
Let m2=t
t28t9=0
t=m2=9,  1( not possible)
m=±3

y=3x+1y=3x1
Therefore point of intersection of common tangents  P(13,0)
Let P divides SS in a ratio of m:n
P(13, 0)=P(3m+3nm+n, 0) 
mn=9m+9nmn=54

 

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