Question

A variable tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ meets the transverse axis and the tangent at the vertex (a, 0) at Q and R. If the locus of mid point of QR is

lx (ky² + mb²) = ab² then value of k² – (l + m)² is equal to-

• A. 0
• B. 6
• C. 12
• D. None of these

Answer 1

Answer: (C)

12

Answer 2

Equation tangent at the point q is $\frac{x}{a}$sec q – $\frac{y}{b}$tan q = 1. It meets the transverse axis y = 0 and the tangent at vertex x = a in the point Q (a cos q, 0) and R $\left\lbrack a,\frac{b(1 - \cos\theta)}{\sin\theta} \right\rbrack$. If (h, k) is mid point of QR then

2h = a + a cos q

Ž cos q = $\frac{2h - a}{a}$ and 2k =$\frac{b(1 - \cos\theta)}{\sin\theta}$

and 4hk = $\frac{ab(1 - \cos^{2}\theta)}{\sin\theta}$ = ab sin q

Ž 16h²k² = a²b² (1 – cos² q)

= a²b² $\left\lbrack 1 - \left( \frac{2h - a}{a} \right)^{2} \right\rbrack$

Ž 4hk² + b²h = b²a locus x(4y² + b²) = ab²

Ž l = 1, k = 4, m = 1

\ k² – (l + m)² = (4)² – (1 + 1)² = 16 – 4 = 12.