Question

Tangents drawn from (c, d) to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ make angles $\alpha$ and $\beta$ with the x - axis. If tan$\alpha.\tan\beta = 1$ then c² - d² =

• A. a² - b²
• B. b² - a²
• C. a² + b²
• D. a²b²

Answer 1

Answer: (C)

a² + b²

Answer 2

If ‘m’ is the slope of the tangent through (c, d) to the Hyperbola, then

(e² – a²) m² – 2cdm+d²+b² =0

⇒ m₁m₂ = $\frac{d^{2} + b^{2}}{c^{2} - a^{2}}$

⇒ tanα. tanβ = $\frac{d^{2} + b^{2}}{c^{2} - a^{2}} = 1$

⇒ e² – e² = a² +b²