Question

P is a point on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, N is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT, ON is equal to

• A. e²
• B. a²
• C. b²
• D. b²/a²

Answer 1

Answer: (B)

a²

Answer 2

Let P(x₁, y₁) be a point on the hyperbola. Then the coordinates of N are (x₁, 0). The equation of the tangent at (x₁, y₁) is $\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1$.

This meets x-axis at T $\left( \frac{a^{2}}{x_{1}},0 \right)$

∴ OT . ON = $\frac{a^{2}}{x_{1}}xx_{1} = a^{2}$.