Question

A normal to the hyperbola $\frac{x^{2}}{a^{2}}$ – $\frac{y^{2}}{b^{2}}$ = 1 meets the axis in A and B the line AL and BL are drawn at Right angle to the axis and meet at L then locus of L is –

• A. a²x² – b²y² = (a² + b²)²
• B. b²x² – a²y² = (a² + b²)²
• C. 4(a²x² – b²y²) = (a² + b²)²
• D. None of these

Answer 1

Answer: (A)

a²x² – b²y² = (a² + b²)²

Answer 2

Normal to the hyperbola is

ax cos q + by cot q = a² + b²

Putting y = 0 and x = 0

We get A$\left( \frac{a^{2} + b^{2}}{a}\sec\theta,0 \right)$, B$\left( 0,\frac{a^{2} + b^{2}}{b}\tan\theta \right)$

line through A, perpendicular to the x-axis is

x = $\frac{a^{2} + b^{2}}{a}$ sec q

line through B, perpendicular to the y-axis is y

= $\frac{a^{2} + b^{2}}{a}$ tan q

So intersection of these point is L

Q sec² q – tan² q = 1

Ž a²x² – b²y² = (a² + b²)².