Question

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

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

Answer 1

Answer: (B)

Given, $ x=3(cos\text{ }t+sin\text{ }t) $
$ \Rightarrow $ $ \frac{x}{3}=\cos t+\sin t $ ...(i) and $ y=4(cos\text{ }t-sin\text{ }t) $
$ \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.