Eliminate the Parameter
-An ellipse: $x = 5 + 2 \cos t , y = 1 + 3 \sin t$
A) $\frac { ( x - 5 ) ^ { 2 } } { 4 } + \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$
B) $\frac { ( x + 5 ) ^ { 2 } } { 4 } + \frac { ( y + 1 ) ^ { 2 } } { 9 } = 1$
C) $\frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1$
D) $( x - 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$

Question

Quizplus · Accepted Answer

The given parametric equations $x = 5 + 2 \cos t$ and $y = 1 + 3 \sin t$ describe an ellipse centered at (5, 1) with semi-major axis lengths of 2 and 3 along the x and y directions, respectively. By isolating $\cos t$ and $\sin t$, and using the identity $\sin^2 t + \cos^2 t = 1$, we can derive the standard form of an ellipse equation, which matches choice A: $\frac { ( x - 5 ) ^ { 2 } } { 4 } + \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$.

Eliminate the Parameter -An Ellipse x=5+2cos⁡t,y=1+3sin⁡tx = 5 + 2 \cos t , y = 1 + 3 \sin tx=5+2cost,y=1+3sint

Eliminate the Parameter
-An Ellipse $x = 5 + 2 \cos t , y = 1 + 3 \sin t$