Find the center, foci, and vertices of the ellipse.
-$$16 ( \mathrm { x } + 2 ) ^ { 2 } + 9 ( \mathrm { y } - 1 ) ^ { 2 } = 144$$
A) center at $( - 1,1 )$
foci at $( - 1,1 - \sqrt { 7 } ) , ( - 1,1 + \sqrt { 7 } )$
vertices at $( - 1,5 ) , ( - 1 , - 3 )$

B) center at $( 1 , - 2 )$
foci at $( 1 , - 2 - \sqrt { 7 } ) , ( 1 , - 2 + \sqrt { 7 } )$
vertices at $( 1,5 ) , ( 1 , - 3 )$

C) center at $( 2,1 )$
foci at $( 2,1 - \sqrt { 7 } ) , ( 2,1 + \sqrt { 7 } )$
vertices at $( 2,5 ) , ( 2 , - 3 )$

D) center at $( - 2,1 )$
foci at $( - 2,1 - \sqrt { 7 } ) , ( - 2,1 + \sqrt { 7 } )$
vertices at $( - 2,5 ) , ( - 2 , - 3 )$

Question

Find the center, foci, and vertices of the ellipse.
-$$16 ( \mathrm { x } + 2 ) ^ { 2 } + 9 ( \mathrm { y } - 1 ) ^ { 2 } = 144$$
A) center at $( - 1,1 )$
foci at $( - 1,1 - \sqrt { 7 } ) , ( - 1,1 + \sqrt { 7 } )$
vertices at $( - 1,5 ) , ( - 1 , - 3 )$

B) center at $( 1 , - 2 )$
foci at $( 1 , - 2 - \sqrt { 7 } ) , ( 1 , - 2 + \sqrt { 7 } )$
vertices at $( 1,5 ) , ( 1 , - 3 )$

C) center at $( 2,1 )$
foci at $( 2,1 - \sqrt { 7 } ) , ( 2,1 + \sqrt { 7 } )$
vertices at $( 2,5 ) , ( 2 , - 3 )$

D) center at $( - 2,1 )$
foci at $( - 2,1 - \sqrt { 7 } ) , ( - 2,1 + \sqrt { 7 } )$
vertices at $( - 2,5 ) , ( - 2 , - 3 )$

Quizplus · Accepted Answer

The equation of the ellipse is given in the form $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $(h, k)$ is the center, $a$ and $b$ are the semi-major and semi-minor axes, respectively. Dividing the given equation by 144 gives $\frac{(x+2)^2}{9} + \frac{(y-1)^2}{16} = 1$, indicating the center is at $(-2, 1)$, with $a^2 = 16$ and $b^2 = 9$. The foci are calculated using $c^2 = a^2 - b^2$, which gives $c = \sqrt{7}$, and since $a > b$, the ellipse is vertical, placing the foci and vertices along the y-axis from the center.

Find the Center, Foci, and Vertices of the Ellipse 16(x+2)2+9(y−1)2=14416 ( \mathrm { x } + 2 ) ^ { 2 } + 9 ( \mathrm { y } - 1 ) ^ { 2 } = 14416(x+2)2+9(y−1)2=144

Find the Center, Foci, and Vertices of the Ellipse $16 ( \mathrm { x } + 2 ) ^ { 2 } + 9 ( \mathrm { y } - 1 ) ^ { 2 } = 144$