Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
-$$\frac { ( y - 1 ) ^ { 2 } } { 4 } - \frac { ( x - 3 ) ^ { 2 } } { 49 } = 1$$
A) center: $( 3,1 )$
transverse axis is parallel to $\mathrm { y }$-axis
vertices: $( 3,1 - \sqrt { 53 } )$ and $( 3,1 + \sqrt { 53 } )$;
foci: $( 3 , - 1 )$ and $( 3,3 )$
asymptotes of $y - 1 = - \frac { 7 } { 2 } ( x - 3 )$ and $y - 1 = \frac { 7 } { 2 } ( x - 3 )$
B) center: $( 3,1 )$
transverse axis is parallel to $\mathrm { y }$-axis
vertices: $( 1,0 )$ and $( 4,4 )$
foci: $( 1,2 - \sqrt { 53 } )$ and $( 4,2 + \sqrt { 53 } )$
asymptotes of $y - 1 = - \frac { 7 } { 2 } ( x - 3 )$ and $y - 1 = \frac { 7 } { 2 } ( x - 3 )$
C) center: $( 3,1 )$
transverse axis is parallel to $y$-axis
vertices: $( 3 , - 1 )$ and $( 3,3 )$
foci: $( 3,1 - \sqrt { 53 } )$ and $( 3,1 + \sqrt { 53 } )$
asymptotes of $\mathrm { y } - 1 = - \frac { 2 } { 7 } ( x - 3 )$ and $y - 1 = \frac { 2 } { 7 } ( x - 3 )$
D) center: $( - 3 , - 1 )$
transverse axis is parallel to $x$-axis
vertices: $( - 3 , - 3 )$ and $( - 3,1 )$
foci: $( - 3 , - 1 - \sqrt { 53 } )$ and $( - 3 , - 1 + \sqrt { 53 } )$
asymptotes of $y - 1 = - \frac { 2 } { 7 } ( x - 3 )$ and $y - 1 = \frac { 2 } { 7 } ( x - 3 )$

Question

Quizplus · Accepted Answer

The given equation is in the standard form of a vertical hyperbola $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$, where the center is $(h, k) = (3, 1)$. The transverse axis is parallel to the y-axis. The vertices are $(3, 1 \pm a) = (3, 1 \pm 2)$, which are $(3, -1)$ and $(3, 3)$. The foci are $(3, 1 \pm c)$, where $c = \sqrt{a^2 + b^2} = \sqrt{4 + 49} = \sqrt{53}$, so the foci are $(3, 1 \pm \sqrt{53})$. The asymptotes are given by the equations $y - 1 = \pm \frac{a}{b}(x - 3)$, which simplifies to $y - 1 = \pm \frac{2}{7}(x - 3)$.

Find the Center, Transverse Axis, Vertices, Foci, and Asymptotes of the Hyperbola