Question

The curve represented by $x=3(\cos t+\sin t)$, $y=4(\cos t-\sin t)$ is

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

Answer 1

Answer: (A)

Given relations are
$x=3(\cos t+\sin t), y=4(\cos t-\sin t)$
$\Rightarrow \frac{x}{3}=\cos t+\sin t, \frac{y}{4}=\text{cos t}-\sin t$
$\Rightarrow \left(\frac{x}{3}\right)^{2}+\left(\frac{y}{4}\right)^{2}=(\cos t+\sin t)^{2}$
$+(\cos t-\sin t)^{2}$
$\Rightarrow \frac{x^{2}}{9}+\frac{y^{2}}{16}=2$
$\Rightarrow \frac{x^{2}}{18}+\frac{y^{2}}{32}=1$, which is an ellipse.