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 :

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 : (A) 5: 4 (B) 14: 13 (C) 2: 1 (D) 13: 1 1

Tardigrade Admin · Accepted Answer

Equation of tangents y2 = 12x ⇒ y = 2x +(3/m) (x2/1) - (y2/8) = 1 ⇒ y =mx ± √m2 -8 ∴ (3/m) = ± √m2 -8 m4 -8m2 -9 = 0 m = ±3 Or (x2/1) - (y2/8) = 1 e = 3 ae = 3

Q. 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 :

$5 : 4$

$14 : 13$

$2 : 1$

$13 : 11$

Equation of tangents
$y^{2} = 12 x \Rightarrow y = 2 x + \frac{3}{m}$
$\frac{x ^{2}}{1} - \frac{y ^{2}}{8} = 1 \Rightarrow y = m x \pm m^{2} - 8$
$∴ \frac{3}{m} = \pm m^{2} - 8$
$m^{4} - 8 m^{2} - 9 = 0$
$m = \pm 3$
Or $\frac{x ^{2}}{1} - \frac{y ^{2}}{8} = 1$
$e = 3$
$a e = 3$

Q. 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 :

5:4

14:13

2:1

13:11