Question

A line parallel to the y-axis intersects the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1 and its conjugate hyperbola at P and Q respectively. Then the normals at P and Q to the respective curves meet on –

• A. y-axis
• B. x-axis
• C. asymptote
• D. None of these

Answer 1

Answer: (B)

x-axis

Answer 2

Let the line parallel to y-axis is x = a, then points are p ŗ $\left( \alpha,\frac{b}{a}\sqrt{\alpha^{2} - a^{2}} \right)$

and Q = $\left( \alpha,\frac{b}{a}\sqrt{\alpha^{2} + a^{2}} \right)$.

Equation of normals are y – $\sqrt{\alpha^{2} - a^{2}}$= $\frac{- a^{2}}{b^{2}}$

$\sqrt{\alpha^{2} - a^{2}}$ (x – a) and

y –$\sqrt{\alpha^{2} + a^{2}}$= $\frac{- a^{2}}{b^{2}}$ $\sqrt{\alpha^{2} + a^{2}}$ (x –a)

Now putting y = 0 in both the normal we get

x = $\left( \frac{b^{2}}{a^{2}} + 1 \right)$a.

So, they intersect on x-axis. Hence (2) is correct answer.