A) a parabola
B) an ellipse
C) a hyperbola
D) None of these
Correct Answer: B
Solution :
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.
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