Question

The curve described parametrically by $x=5\cos t$ and $y=2\sin t$ represents

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

Answer 1

The correct option is C An ellipse

$x=5\cos t,y=2\sin t$

$\cos t=\frac{x}{5}\cdots \cdots \left(1\right)$
$\sin t=\frac{y}{2}\cdots \cdots \left(2\right)$

Squaring $\left(1\right)$ and $\left(2\right)$ and then adding both

${\left(\frac{x}{5}\right)}^2+{\left(\frac{y}{2}\right)}^2=1$
$\frac{x^2}{25}+\frac{y^2}{4}=1$
$4x^2+25y^2-100=0$
On comparing with

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

We have $a=4,b=25,h=0,g=0,f=0,c=-100$

$\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2=-10000\ne 0$
and $h^2<ab$
So, its an ellipse.