Question

If a chord of a rectangular hyperbola, parallel to its conjugate axis subtends angles θ₁ and θ₂ at its vertices, then

• A. θ₁ + θ₂ = $\frac{\pi}{2}$
• B. θ₁ + θ₂ = π
• C. θ₁ + θ₂ = $\frac{3\pi}{4}$
• D. None of these

Answer 1

Answer: (B)

θ₁ + θ₂ = π

Answer 2

Let the hyperbola be x² – y² = a² and the chord be x = k. It meet the curve at $(k,\sqrt{k^{2} - a^{2}})$ and $(k, - \sqrt{k^{2} - a^{2}})$.

Hence tanθ₁ =$\frac{\frac{\sqrt{k^{2} - a^{2}}}{k - a} + \frac{\sqrt{k^{2} - a^{2}}}{k - a}}{1 - \frac{k^{2} - a^{2}}{(k - a)^{2}}} = - \frac{1}{a}\sqrt{k^{2} - a^{2}}$.

Also tan θ₂ =$\frac{1}{a}\sqrt{k^{2} - a^{2}}$ = –tan θ₁ = tan (π – θ₁)

⇒ θ₁ + θ₂ = π