JEE PYQ: Hyperbola Question 15
Question 15 - 2019 (12 Apr Shift 1)
Let P be the point of intersection of the common tangents to the parabola $y^2 = 12x$ and hyperbola $8x^2 - y^2 = 8$. If S and S’ denote the foci of the hyperbola where S lies on the positive $x$-axis then P divides SS’ in a ratio:
(1) 13 : 11
(2) 14 : 13
(3) 5 : 4
(4) 2 : 1
Show Answer
Answer: (3)
Solution
Tangent to parabola: $y = mx + \frac{3}{m}$. Tangent to hyperbola $\frac{x^2}{1} - \frac{y^2}{8} = 1$: $c^2 = m^2 - 8$. So $\frac{9}{m^2} = m^2 - 8$, i.e. $m^4 - 8m^2 - 9 = 0$, $(m^2+1)(m^2-9) = 0$, $m = \pm 3$. Tangent: $y = 3x + 1$ or $y = -3x - 1$. Intersection $P\left(-\frac{1}{3}, 0\right)$. $e = \sqrt{1+8} = 3$, foci $S(3, 0)$ and $S’(-3, 0)$. $\frac{SP}{S’P} = \frac{3+\frac{1}{3}}{3-\frac{1}{3}} = \frac{10/3}{8/3} = \frac{5}{4}$.
BETA
