If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix.
-$$4 x ^ { 2 } + 9 y ^ { 2 } + 80 x - 108 y + 688 = 0$$
A) C: (-10, 6); V: (-10, 9), (-10, 3); F: (-10, 8.2), (-10, 3.8)
B) C: (-10, 6); V: (-7, 6), (-13, 6); F: (-7.8, 6), (-12.2, 6)
C) C: (-10, 6); V: (-8, 6), (-12, 6); F: (-6.4, 6), (-13.6, 6)
D) C: (-10,6); V: (-10, 8), (-10, 4); F: (-10, 9.6), (-10, 2.4)

Question

Quizplus · Accepted Answer

We can rewrite the given equation as 
$$\frac{(x+10)^2}{25}-\frac{(y-6)^2}{16}=1$$
This is the equation of an ellipse with center $C=(-10,6)$, horizontal semi-major axis of length $5$ and vertical semi-minor axis of length $4$. Therefore, the vertices are $V_1=(-7,6)$ and $V_2=(-13,6)$, and the foci are $F_1=(-7.8,6)$ and $F_2=(-12.2,6)$.

If the Equation Represents a Hyperbola, Find the Center, Foci 4x2+9y2+80x−108y+688=04 x ^ { 2 } + 9 y ^ { 2 } + 80 x - 108 y + 688 = 04x2+9y2+80x−108y+688=0

If the Equation Represents a Hyperbola, Find the Center, Foci $4 x ^ { 2 } + 9 y ^ { 2 } + 80 x - 108 y + 688 = 0$