Let P(3 sec θ, 2 tan θ) and Q(3 sec φ, 2 tan φ) where θ+φ=(π/2), be two distinct points on the hyperbola (x2/9)-(y2/4)=1. Then the ordinate of point of intersection of the normals at P and Q is

Question

Let P(3 sec θ, 2 tan θ) and Q(3 sec φ, 2 tan φ) where θ+φ=(π/2), be two distinct points on the hyperbola (x2/9)-(y2/4)=1. Then the ordinate of point of intersection of the normals at P and Q is (A) (11/3) (B) (-11/3) (C) (13/2) (D) (-13/2)

Tardigrade Admin · Accepted Answer

We have, 3 x cos θ+2 y cot θ=13 .....(1) 3 x sin θ+2 y tan θ=13 ....(2) ∴ On solving (1) and (2), we get y=(-13/2)

Q. Let $P (3 sec θ, 2 tan θ)$ and $Q (3 sec ϕ, 2 tan ϕ)$ where $θ + ϕ = \frac{π}{2}$ , be two distinct points on the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ . Then the ordinate of point of intersection of the normals at $P$ and $Q$ is

$\frac{11}{3}$

$\frac{- 11}{3}$

$\frac{13}{2}$

$\frac{- 13}{2}$

We have,
$3 x cos θ + 2 y cot θ = 13$ .....(1)
$3 x sin θ + 2 y tan θ = 13$ ....(2)
$∴$ On solving (1) and (2), we get
$y = \frac{- 13}{2}$

Q. Let P(3secθ,2tanθ) and Q(3secϕ,2tanϕ) where θ+ϕ=2π​, be two distinct points on the hyperbola 9x2​−4y2​=1. Then the ordinate of point of intersection of the normals at P and Q is

311​

3−11​

213​

2−13​