Question

Chord of contact is drawn from point $P$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ which forms a triangle of constant area with the coordinate axes. Then the locus of point $P$ is

• A. $xy=c^2$
• B. $x+y=c^2$
• C. $x^2+y^2=c^2$
• D. $x+\frac{1}{y}=c^2$

Answer 1

The correct option is A $xy=c^2$

Let $P\equiv (x_1,y_1)$
The chord of contact from $P$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $T=0$
$\Rightarrow \frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=1$
It meets the coordinate axis at the points $Q(\frac{a^2}{x_1},0)$ and $R(0,\frac{b^2}{y_1}).$
Area of $△PQR=\frac{1}{2}\frac{a^2{b}^2}{x_1{y}_1}=c_1$ (constant)
$\Rightarrow x_1{y}_1=\frac{a^2{b}^2}{2c_1}=c^2$
(where $c$ is another constant)
Hence required locus is
$\Rightarrow xy=c^2.$