Question

The curve with parametric equations $x=3(\cos t+\sin t)$ , $y=4(\cos t-\sin t)$ is

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

Answer 1

Answer: (A)

Ellipse

Given, $x=3(\cos t+\sin t)$

$\Rightarrow \frac{x}{3}=\cos t+\sin t$

$\Rightarrow \frac{x^2}{9}={\cos }^{2}t+{\sin }^{2}t+2\cos t..\sin t$ ...(i)

And $y=4(\cos t-\sin t)$

$\Rightarrow \frac{y}{4}=\cos t-\sin t$

$\Rightarrow \frac{y^2}{16}={\cos }^{2}t+{\sin }^2-2\cos t\sin t$ ...(ii)

Adding (i) and (ii) gives us

$\frac{x^2}{4}+\frac{y^2}{6}=2({\cos }^{2}t+{\sin }^{2}t)=2$

Hence, $\frac{x^2}{4}+\frac{y^2}{6}=2$

This represents an ellipse.