Let $P$ be the point of intersection of the common tangents to the parabola $y^{2}=12 x$ and the hyperbola $8 x^{2}-y^{2}=8$. If $S$ and $S^{\prime}$ denote the foci of the hyperbola where $S$ lies on the positive $x$-axis then $P$ divides $S S^{\prime}$ in a ratio:

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Hyperbola

Let

P

be the point of intersection of the common tangents to the parabola

y^{2} = 12 x

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

S S^{'}

in a ratio:

Answer: (C)

A tangent to

y^{2} = 12 x

has the form

y = m x + \frac{3}{m}

, while a tangent to

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

has the form

y = m x \pm m^{2} - 8

. For a common tangent the intercepts match:

\frac{3}{m} = \pm m^{2} - 8

, so

\frac{9}{m ^{2}} = m^{2} - 8

and

m^{4} - 8 m^{2} - 9 = 0

, giving

m = \pm 3

. Taking

y = 3 x + 1

, it meets the

x

-axis at

P (- \frac{1}{3}, 0)

. The hyperbola has

a = 1

and

e = 3

, so its foci are

S (3, 0)

and

S^{'} (- 3, 0)

. Then

P S = \frac{10}{3}

and

P S^{'} = \frac{8}{3}

, so

P

divides

S S^{'}

in the ratio

5 : 4

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option