Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the…

Question

Let P be the point of intersection of the common tangents to the parabola y2
 = 12x and the hyperbola
8x2
-- y2
 = 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 :

Accepted Answer

Let equation of common tangent is y = mx + $${3 \over m}$$

$$ \therefore $$ $${\left( {{3 \over m}} \right)^2}$$ = 1, m2 - 8

$$ \Rightarrow {m^4} - 8{m^2} - 9 = 0$$

$$ \Rightarrow {m^2} = 9 \Rightarrow m =  \pm 3$$

$$ \therefore $$ equation of common tangents are y = 3x + 1 \& y = -3x - 1

$$ \therefore $$ $${{PS} \over {PS}} = {{3 + {1 \over 3}} \over { - {1 \over 3} + 3}} = {5 \over 4}$$

Solutions · 1