Question

The locus of point of intersection of the lines $\frac{tx}{a}-\frac{y}{b}+t=0,\frac{x}{a}+\frac{ty}{b}-1=0$ is ( $t$ is a parameter)

• A. parabola
• B. circle
• C. hyperbola
• D. ellipse

Answer 1

The correct option is D ellipse

Given, $\frac{tx}{a}-\frac{y}{b}+t=0\Rightarrow t=\frac{y/b}{x/a+1}$

and $\frac{x}{a}+\frac{ty}{b}-1=0$

Now eliminating $t$ from the above equation we get,

$\frac{x}{a}+\frac{y}{b}\times \frac{ay}{b(x+a)}-1=0$

$\Rightarrow x{b}^2(x+a)+a^2{y}^2=a{b}^2(x+a)$

$\Rightarrow x^2{b}^2+a^2{y}^2=a^2{b}^2$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, which is clearly an ellipse.