Question

The curve described parametrically by $x=\sin {}^{2}t,y=2\cos t$ represents

• A. Circle
• B. Parabola
• C. An elipse
• D. Hyperbola

Answer 1

The correct option is B Parabola

$x=\sin {}^{2}t,y=2\cos t$

On Squaring $y$

$\Rightarrow y^2=4{\cos }^{2}t$
$\Rightarrow {\cos }^{2}t=\frac{y^2}{4}\cdots \cdots \left(1\right)$
${\sin }^{2}t=x\cdots \cdots \left(2\right)$

Adding $\left(1\right)$and $\left(2\right)$

$x+{\left(\frac{y}{2}\right)}^2=1$
$4x+y^2-4=0$

On comparing with

$a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$

We have , $a=0,b=1,h=0,g=2,f=0,c=-4$

$\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2=-4$
and $h^2=ab$

So, it represents a parabola