Question

The locus of the point of intersection of the lines $x =a\left(\frac{1-t^{2}}{1+t^{2}}\right) $ and $ y = \frac{2at }{1+t^{2}} $ represent ($t$ being a parameter)

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

Answer 1

Answer: (A)

To eliminate the parameter $t$, square and add the equations, we have
$x^{2} +y^{2} =a^{2} \left(\frac{1-t^{2}}{1+t^{2}}\right)^{2} + \frac{4a^{2}t^{2}}{\left(1+t^{2}\right)^{2}} $
$ = \frac{a^{2}}{\left(1+t^{2}\right)^{2}} \left[\left(1+t^{2}\right)^{2} +4t^{2}\right] $
$= \frac{a^{2} \left(1+t^{2}\right)^{2}}{\left(1+t^{2}\right)^{2}} =a^{2} $
Which is the equation of a circle.