The correct option is C 5:4
Equation of parabola y2=12x
So equation of its tangent : y=3x+3m
Equation of hyperbola x21−y28=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±√m2−8
Both tangent are coomon tangents
Therefore, 9m2=m2−8
Let m2=t
t2−8t−9=0
⇒t=m2=9, −1( not possible)
⇒m=±3
∴y=3x+1y=−3x−1
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)
⇒−m−n=−9m+9n⇒mn=54