Question

The curve described parametrically by $y=2at$ and $x=a{t}^2$ represents

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

Answer 1

The correct option is B a parabola

$x=a{t}^2,y=2at$
Eliminating the parameter $t,$
$t=\frac{y}{2a}\Rightarrow x=a{\left(\frac{y}{2a}\right)}^2\Rightarrow y^2=4ax\Rightarrow y^2-4ax=0$

Comparing with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$
$a=0,b=1,c=f=h=0,g=-2a$
$\therefore △=\left|\begin{array}{ccc}0& 0& -2a\\ 0& 1& 0\\ -2a& 0& 0\end{array}\right|=-4a^2\ne 0$

Also, $h^2-ab=0$
Hence, given curve represents a parabola.