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.
-$$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 13 = 25$$
A) (2, -3); r = 25
B) (3, -2); r = 5
C) (-2, 3); r = 5
D) (-3, 2); r = 25

Question

Quizplus · Accepted Answer

The given equation is of a circle, as it can be rewritten in the standard form $(x-h)^2 + (y-k)^2 = r^2$ after completing the square. Completing the square for both $x$ and $y$ terms: $$(x^2 - 6x + 9) + (y^2 + 4y + 4) = 25 + 9 + 4$$ simplifies to $$(x-3)^2 + (y+2)^2 = 38$$. However, there seems to be a mistake in my calculation or interpretation since none of the options match the correct radius based on the calculation. The correct center is indeed (3, -2), but the radius calculation should be revisited. Correcting my explanation: the equation simplifies to $(x-3)^2 + (y+2)^2 = 36$, making the radius $r = 6$, not directly matching any of the provided options. The correct approach identifies the center correctly as (3, -2), but the radius provided in the options and my initial explanation do not align with the standard form equation derived.

If the Equation Represents a Hyperbola, Find the Center, Foci x2+y2−6x+4y+13=25x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 13 = 25x2+y2−6x+4y+13=25

If the Equation Represents a Hyperbola, Find the Center, Foci $x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 13 = 25$