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.