Question

If the polars of (x₁, y₁) and (x₂, y₂) w.r.t. the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ are at right angles, then $\frac{x_{1}x_{2}}{y_{1}y_{2}} =$

• A. $\frac{- a^{2}}{b^{2}}$
• B. $\frac{a^{2}}{b^{2}}$
• C. $\frac{a^{4}}{b^{4}}$
• D. $- \frac{a^{4}}{b^{4}}$

Answer 1

Answer: (D)

$- \frac{a^{4}}{b^{4}}$

Answer 2

Polar of (x₁, y₁) w.r.t. the given hyperbola is $\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1$.

Its slope = $\frac{x_{1}/a^{2}}{y_{1}/b^{2}} = \frac{b^{2}x_{1}}{a^{2}y_{1}}$.

Similarly, slope of polar of (x₂, y₂) w.r.t. the given hyperbola is $\frac{b^{2}x_{2}}{a^{2}y_{2}}$.

$\because$Polars are ⊥, ∴ $\frac{b^{2}x_{1}}{a^{2}y_{1}}.\frac{b^{2}x_{2}}{a^{2}y_{2}} = - 1$ (or) $\frac{x_{1}x_{2}}{y_{1}y_{2}} = \frac{- a^{4}}{b^{4}}$.