Question

The locus of the point, tangents from which to the rectangular hyperbola x² – y² = a² contain an angle of 45° is

• A. (x² + y²) + a²(x² – y²) = 4a⁴
• B. 2(x² + y²) + 4a²(x² – y²) = 4a⁴
• C. (x² + y²) + 4a²(x² – y²) = 4a⁴
• D. (x² + y²) + a²(x² – y²) = a⁴

Answer 1

Answer: (C)

(x² + y²) + 4a²(x² – y²) = 4a⁴

Answer 2

Let y = mx ± $\sqrt{m^{2}a^{2} - a^{2}}$ be two tangents and passing through (h, k). Then

(k – mk)² = m²a² – a² ⇒ m²(h² – a²) – 2khm + k² + a² = 0.

⇒ m₁ + m₂ = $\frac{2kh}{h^{2} - a^{2}}$ and m₁m₂ =$\frac{k^{2} + a^{2}}{h^{2} - a^{2}}$, and tan45⁰ =$\frac{m_{1} + m_{2}}{1 + m_{1}m_{2}}$