The ellipse shown in the figure has centre at the origin, major axis along the $x$-axis, and vertices at $(\pm a,0)$. Which parametric point on this ellipse corresponds to eccentric angle $\theta$? 
A$(a\cos\theta,\, b\sin\theta)$
B$(a\sin\theta,\, b\cos\theta)$
C$(b\cos\theta,\, a\sin\theta)$
D$(a\cos\theta,\, a\sin\theta)$
Answer & Solution
Correct answer: A. $(a\cos\theta,\, b\sin\theta)$
For the standard ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the usual parametric representation is $x=a\cos\theta$, $y=b\sin\theta$. Substituting gives $\frac{a^2\cos^2\theta}{a^2}+\frac{b^2\sin^2\theta}{b^2}=\cos^2\theta+\sin^2\theta=1$. The figure of the auxiliary circle also supports this correspondence.
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