Question

The curve described parametrically by $x=3\cos \left(2t\right),y=1+{\cos }^2\left(2t\right)$ represents

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

Answer 1

The correct option is B Parabola

$x=3\cos \left(2t\right),y=1+{\cos }^2\left(2t\right)$

${\cos }^2\left(2t\right)={\left(\frac{x}{3}\right)}^2$
${\cos }^2\left(2t\right)=\frac{x^2}{9}\cdots \cdots \left(1\right)$

${\cos }^2\left(2t\right)=y-1\cdots \cdots \left(2\right)$

Equalising equation $\left(1\right)$ and $\left(2\right)$

$y-1=\frac{x^2}{9}$
$x^2-9y+9=0$

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

We have $a=1,b=0,h=0,g=0,f=-\frac{9}{2},c=9$

$\Delta =0+0-\frac{81}{4}-0-0\ne 0$
and $h^2=ab$

Hence it represents a parabola .