Question

Locus of mid-points of chords of the hyperbola

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, such angle between tangents at the end of the chords is 90⁰, is:

• A. $x^{2} + y^{2} = \left( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} \right)^{2}$
• B. $(x^{2} - y^{2}) = \left( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} \right)^{2}$
• C. $x^{2} + y^{2} = (a^{2} - b^{2})\left( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} \right)^{2}$
• D. $x^{2} + y^{2} = (a^{2} - b^{2})^{2}\left( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} \right)^{2}$

Answer 1

Answer: (C)

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

Answer 2

Tangents are drawn from its director circle

x² + y² = a² - b².

General point on it $(\sqrt{a^{2} - b^{2}}$ cos θ, $\sqrt{a^{2} - b^{2}}$ sin θ).

Chord of contact is T = 0

⇒ $\frac{\sqrt{a^{2} - b^{2}}x\cos\theta}{a^{2}} - \frac{\sqrt{a^{2} - b^{2}}y\sin\theta}{b^{2}} = 1$Compare it

with T = S₁

⇒ $\frac{hx}{a^{2}} - \frac{ky}{b^{2}} = \frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}}$

Now ⇒ sin²θ + cos²θ = 1

$\frac{k^{2}}{\left( \frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}} \right)^{2}} + \frac{h^{2}}{\left( \frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}} \right)^{2}} = (a^{2} - b^{2})$.