Question

The curve described parametrically by $x=2(\cos t+\sin t),y=5(\cos t-\sin t)$ represents

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

Answer 1

The correct option is C an ellipse

Given, $x=2(\cos t+\sin t)$
$\Rightarrow \frac{x}{2}=(\cos t+\sin t)\cdots \left(i\right)$
and $y=5(\cos t-\sin t)$
$\Rightarrow \frac{y}{5}=(\cos t-\sin t)\cdots \left(ii\right)$

From $\left(i\right)$ and $\left(ii\right),$
${\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{5}\right)}^2$
$=(\cos t+\sin t{)}^2+(\cos t-\sin t{)}^2$
$={\cos }^{2}t+{\sin }^{2}t+2\cos t\sin t+{\cos }^{2}t+{\sin }^{2}t-2\cos t\sin t$
$=2({\cos }^{2}t+{\sin }^{2}t)$
$=2\times 1=2$
$\Rightarrow \frac{x^2}{4}+\frac{y^2}{25}=2$
$\Rightarrow 25x^2+4y^2-200=0$

On comparing with $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0,$ we have
$a=25,b=4,h=0,g=0,f=0,c=-200$
Then $\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2$
$=25\times 4\times (-200)+0-0-0-0$
$=-20000\ne 0\cdots \left(iii\right)$
Now, $h^2-ab=0-100=-100<0\cdots \left(iv\right)$
From $\left(iii\right)$ and $\left(iv\right)$, the above equation represents an ellipse.