Question

The curve represented by, $x=2(\cos t+\sin$$t)$ and $y=5(\cos t-\sin t)$ is

• A. Circle
• B. Parabola
• C. Ellipse
• D. Hyperbola

Answer 1

Answer: (C)

Ellipse

Given, $x=2(\cos t+\sin t)$

$\Rightarrow \frac{x}{2}=\cos t+\sin t$ ...(i)

And $y=5(\cos t-\sin t)$

$\Rightarrow \frac{y}{5}=\cos t-\sin t$ ...(ii)

Then ${\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{5}\right)}^2$

$={(\cos t+\sin t)}^2+{(\cos t-\sin t)}^2$

$={\cos }^{2}t+{\sin }^{2}t+2\cos t.\sin t+{\cos }^{2}t+{\sin }^{2}t-2\cos t.\sin t$

$=2({\cos }^{2}t+{\sin }^{2}t)$

$=2$

Hence, ${\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{5}\right)}^2=2$

$\frac{x^2}{4}+\frac{y^2}{25}=2$

$\frac{x^2}{8}+\frac{y^2}{50}=1$

This represents an ellipse.