Question

Let $t$ being a variable parameter, then prove that the locus of the point $x=a\frac{1-t^2}{1+t^2},y=b\frac{2t}{1+t^2}$ is an ellipse.

Answer 1

Let $x=a\frac{1-t^2}{1+t^2},y=b\frac{2t}{1+t^2}$, for $t$ being variable parameter.

or, $\frac{x}{a}=\frac{1-t^2}{1+t^2}$......(1) and $\frac{y}{b}=\frac{2t}{1+t^2}$......(2).

Now squaring and adding (1) and (2) we get,

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{(1-t^2{)}^2+4t^2}{(1+t^2{)}^2}$

or, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{(1+t^2{)}^2}{(1+t^2{)}^2}$

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

So locus of $(x,y)$ is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, an ellipse.