Question

If the line lx + my + n = 0 meets the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1 at the extremities of a pair of conjugate diameters, then the relation a² – b²m² is equal to –

• A. 1
• B. 2
• C. 0
• D. None of these

Answer 1

Answer: (C)

0

Answer 2

Let CP and CD be a pair of conjugate diameters of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1. Then the co-ordinates of P and D are (a sec q, b tan q) and (a tan q, b sec q) respectively.

It is given that the line lx + my + n = 0 meets the hyperbola at P and D. Therefore,

al sec q + bm tan q + n = 0

and al tan q + bm sec q + n = 0

Ž al sec q + bm tan q + n = –n

and al tan q + bm sec q = – n

Ž (al sec q + bm tan q)² = n² … (i)

And (al tan q + bm sec q)² = n² … (ii)

on subtracting Equation (ii) from (i), we get

a²l² (sec² q – tan² q) + b²m² (tan² q – sec² q) = 0

Ž a²l² – b²m² = 0