Find the equation of the circle $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$ that passes through the points. ?? $( 0,0 ) , ( 0,12 ) , ( 6,6 )$ ?
A)? $x ^ { 2 } - y ^ { 2 } + 12 y = 0$
B)? $x ^ { 2 } - y ^ { 2 } - 12 y = 0$
C)? $- x ^ { 2 } + y ^ { 2 } - 12 y = 0$
D)? $x ^ { 2 } + y ^ { 2 } - 12 y = 0$
E)? $x ^ { 2 } + y ^ { 2 } + 12 y = 0$

Question

Quizplus · Accepted Answer

To find the equation of the circle, we substitute the given points into the general equation $x^2 + y^2 + Dx + Ey + F = 0$ and solve for $D$, $E$, and $F$.1. Substituting $(0,0)$: $0 + 0 + 0 + 0 + F = 0$, so $F = 0$.2. Substituting $(0,12)$: $0 + 144 + 0 + 12E + 0 = 0$, so $12E = -144$, and $E = -12$.3. Substituting $(6,6)$: $36 + 36 + 6D + 6E + 0 = 0$, so $6D + 6E = -72$. We already know $E = -12$, so $6D - 72 = -72$, which doesn't provide new information about $D$ directly, but since the circle's equation does not actually depend on $D$ for these points, we focus on what we've found.With $F = 0$ and $E = -12$, the equation becomes $x^2 + y^2 - 12y = 0$, which matches option D.

Find the Equation of the Circle x2+y2+Dx+Ey+F=0x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0x2+y2+Dx+Ey+F=0

Find the Equation of the Circle $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$