Graph Ellipses Not Centered at the Origin
-$$36 ( x + 3 ) ^ { 2 } + 16 ( y - 2 ) ^ { 2 } = 576$$
A) foci at $( - 3,2 - 2 \sqrt { 5 } )$ and $( - 3,2 + 2 \sqrt { 5 } )$
B) foci at $( 2 , - 3 - 2 \sqrt { 5 } )$ and $( 2 , - 3 + 2 \sqrt { 5 } )$
C) foci at $( 3,2 - 2 \sqrt { 5 } )$ and $( 3,2 + 2 \sqrt { 5 } )$
D) foci at $( - 2,2 - 2 \sqrt { 5 } )$ and $( - 2,2 + 2 \sqrt { 5 } )$

Question

Quizplus · Accepted Answer

The given equation is in the form of an ellipse not centered at the origin. The center of the ellipse is at $(-3, 2)$. The major axis is along the y-axis because the coefficient of $(y-2)^2$ is smaller than the coefficient of $(x+3)^2$. To find the foci, we use the formula $c^2 = a^2 - b^2$, where $a^2$ is the larger denominator and $b^2$ is the smaller denominator. Here, $a^2 = 576/16 = 36$ and $b^2 = 576/36 = 16$, so $c^2 = 36 - 16 = 20$, and $c = 2\sqrt{5}$. The foci are along the y-axis (the major axis), so the foci are at $(-3, 2 \pm 2\sqrt{5})$.

Graph Ellipses Not Centered at the Origin - 36(x+3)2+16(y−2)2=57636 ( x + 3 ) ^ { 2 } + 16 ( y - 2 ) ^ { 2 } = 57636(x+3)2+16(y−2)2=576

Graph Ellipses Not Centered at the Origin
- $36 ( x + 3 ) ^ { 2 } + 16 ( y - 2 ) ^ { 2 } = 576$