Find the vertices and foci of the hyperbola.
-$$\frac { ( y - 4 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 1$$
A) Vertices: $( 8,2 ) , ( 0,2 )$; Foci: $( 9,2 ) , ( - 1,2 )$
B) Vertices: $( 2,8 ) , ( 2,0 )$; Foci: $( 2,9 ) , ( 2 , - 1 )$
C) Vertices: $( 7,2 ) , ( 1,2 )$; Foci: $( 1,2 ) , ( 7,2 )$
D) Vertices: $( 2,7 ) , ( 2,1 )$; Foci: $( 2,1 ) , ( 2,7 )$

Question

Quizplus · Accepted Answer

The given equation is of a hyperbola centered at $(2, 4)$ with a vertical transverse axis. The vertices are found by adding and subtracting the square root of the denominator associated with $y$ (which is $\sqrt{16} = 4$) to the $y$-coordinate of the center, resulting in vertices at $(2, 8)$ and $(2, 0)$. The foci are found by calculating $c = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ and adding and subtracting this value to the $y$-coordinate of the center, resulting in foci at $(2, 9)$ and $(2, -1)$.

Find the Vertices and Foci of the Hyperbola (y−4)216−(x−2)29=1\frac { ( y - 4 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 116(y−4)2​−9(x−2)2​=1

Find the Vertices and Foci of the Hyperbola $\frac { ( y - 4 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 1$