Use the result, "the set of parametric equations for the ellipse is $x = h + a \cos \theta , y = k + b \sin \theta$ " to find a set of parametric equations for the ellipse with vertices $( 3,11 )$ and $( 3,1 )$ and with foci at $( 3,9 )$ and $( 3,3 )$.
A) $x = 4 \sin \theta , y = 5 \cos \theta$
B) $x = 6 + 5 \cos \theta , y = 3 + 4 \sin \theta$
C) $x = 5 \cos \theta , y = 4 \sin \theta$
D) $x = 3 - 5 \cos \theta , y = 6 - 4 \sin \theta$
E) $x = 3 + 5 \cos \theta , y = 6 + 4 \sin \theta$

Question

Quizplus · Accepted Answer

Since the center of the ellipse is at $(3,6)$, we have $h=3$ and $k=6$. The distance from the center to each vertex is 5, so we have $a=5$. The distance from the center to each focus is 3, so we have $c=3$ and $b^2=a^2-c^2=16$. Thus, the set of parametric equations for the ellipse is $x=3+5\cos\theta$ and $y=6+4\sin\theta$, which matches choice E.

Use the Result, "The Set of Parametric Equations for the Ellipse