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 equation chord of contact of tangents from $(x_1,y_1)$ to the ellipse is
$\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=1$
It meets the axes at the points $(\frac{a^2}{x_1},0)$ and $(0,\frac{b^2}{y_1})$.
Area of the triangle is $\frac{1}{2}\frac{a^2}{x_1}\frac{b^2}{y_1}=k\left(constant\right)$
$\Rightarrow x_1{y}_1=\frac{a^2{b}^2}{2k}=c^2$ (c is constant)
$\Rightarrow xy=c^2$ is the required locus.