Question

A series of chords of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, are tangents to the circle described on the line joining the foci as diameter. The locus of their poles w.r.t. the hyperbola is

• A. $\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}} = \frac{1}{a^{2} + b^{2}}$
• B. $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{1}{a^{2} + b^{2}}$
• C. $\frac{x^{2}}{a^{4}} - \frac{y^{2}}{b^{4}} = \frac{1}{a^{2} + b^{2}}$
• D. None of these

Answer 1

Answer: (A)

$\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}} = \frac{1}{a^{2} + b^{2}}$

Answer 2

Equation of the hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$... (1)

Its foci are (ae, 0) and (- ae, 0).

Equation of the circle on the join of foci as diameter is

(x - ae) (x + ae) + (y - 0) (y - 0) = 0

or x² + y² = a²e² ... (2)

Let (x₁, y₁) be the pole of a chord of (1).

Equation of chord i.e. polar of (x₁, y₁) w.r.t. (1) is $\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1$. ... (3)

Since it touches the circle (2), the length of ⊥ from the centre (0, 0) of (2) on (3) is equal to radius ae.

⇒ $\pm \frac{1}{\sqrt{\frac{x_{1}^{2}}{a^{4}} + \frac{y_{1}^{2}}{b^{4}}}} = ae$

or $\frac{x_{1}^{2}}{a^{4}} + \frac{y_{1}^{2}}{b^{4}} = \frac{1}{a^{2}e^{2}} = \frac{1}{a^{2} + b^{2}}.\left\lbrack \because e^{2} = \frac{a^{2} + b^{2}}{a^{2}} \right\rbrack$.

∴ Locus of (x₁, y₁) is $\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}} = \frac{1}{a^{2} + b^{2}}$.