Question

The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, forms a triangle of constant area with the coordinate axes is:

• A. a straight line
• B. a hyperbola
• C. an ellipse
• D. a circle

Answer 1

Answer: (B)

a hyperbola

The chord of contact of tangents from $(h,k)$ is $\frac{xh}{a^2}+\frac{yk}{b^2}=1$.

It meets the axes at point $(\frac{a^2}{h},0)$ and $(0,\frac{b^2}{k})$.

Area of the triangle $=\frac{1}{2}\times \frac{a^2}{h}\times \frac{b^2}{k}=c$ (constant)

$\Rightarrow hk=\frac{a^2{b}^2}{2c}$ (c is constant)

$xy=\frac{a^2{b}^2}{2c}$ is the required locus.