Question

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

• A. a parabola
• B. an ellipse
• C. a hyperbola
• D. None of these

Answer 1

Answer: (B)

Given, $x=3(cost+sint)$
$\Rightarrow$ $\frac{x}{3}=\cos t+\sin t$ ...(i) and $y=4(cost-sint)$
$\Rightarrow$ $\frac{y}{4}=\cos t-\sin t$ ...(ii)
On squaring and adding Eqs. (i) and (ii),
${\left(\frac{x}{3}\right)}^2+{\left(\frac{y}{4}\right)}^2={(\cos t+\sin t)}^2+{(\cos t-\sin t)}^2$
$\Rightarrow$ $\frac{x^2}{9}+\frac{y^2}{16}=1+2\cos t\sin t+1-2\cos t\sin t$
$\Rightarrow$ $\frac{x^2}{9}+\frac{y^2}{16}=2$
$\Rightarrow$ $\frac{x^2}{18}+\frac{y^2}{32}=1$
Which represents an ellipse.