Decide whether or not the equation has a circle as its graph. If it does not, describe the graph.
-$$x ^ { 2 } + y ^ { 2 } - 8 x + 12 y - 29 = 0$$
A) no; the graph is the point $( - 4,6 )$
B) no; the graph is nonexistent
C) yes
D) no; the graph is the point $( 4 , - 6 )$

Question

Quizplus · Accepted Answer

The given equation is of the form $x^2 + y^2 + Dx + Ey + F = 0$, which can represent a circle if it can be rewritten in the standard form $(x - h)^2 + (y - k)^2 = r^2$. Completing the square for both $x$ and $y$ terms: $(x^2 - 8x + 16) + (y^2 + 12y + 36) = 29 + 16 + 36$, simplifies to $(x - 4)^2 + (y + 6)^2 = 81$, which is a circle centered at $(4, -6)$ with radius $9$.

Decide Whether or Not the Equation Has a Circle as Its