Identify the type of graph.
$- x ^ { 2 } - 6 x + 5 y ^ { 2 } + 10 y = 25$
A) The graph is a circle with center $\left( \frac { 5 } { 9 } , - \frac { 2 } { 3 } \right)$
B) The graph is a hyperbola opening to the right and left.
C) The equation has no graph.
D) The graph is an ellipse with center $\left( - \frac { 5 } { 9 } , \frac { 2 } { 3 } \right)$

Question

Quizplus · Accepted Answer

The given equation is of the form $Ax^2 + Cy^2 + Dx + Ey + F = 0$, where $A$ and $C$ have opposite signs, indicating a hyperbola. Since the coefficients of $x^2$ and $y^2$ are $-1$ and $5$ respectively, and they have opposite signs, the graph represents a hyperbola. Specifically, because the $x^2$ term is negative, the hyperbola opens to the right and left.

Identify the Type of Graph −x2−6x+5y2+10y=25- x ^ { 2 } - 6 x + 5 y ^ { 2 } + 10 y = 25−x2−6x+5y2+10y=25

Identify the Type of Graph $- x ^ { 2 } - 6 x + 5 y ^ { 2 } + 10 y = 25$