Find the equation of the circle
$$x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$$
that passes through the points $( - 1,1 ) , ( - 6 , - 4 ) , ( 4 , - 4 )$.
A) $x ^ { 2 } + y ^ { 2 } + 2 x + 8 y - 8 = 0$
B) $x ^ { 2 } + y ^ { 2 } + x + 4 y - 8 = 0$
C) $x ^ { 2 } + y ^ { 2 } + 2 x + 8 y + 42 = 0$
D) $x ^ { 2 } + y ^ { 2 } + 2 x + 8 y - 25 = 0$
E) $x ^ { 2 } + y ^ { 2 } + x + 4 y - 25 = 0$

Question

Quizplus · Accepted Answer

To find the equation of the circle, we can substitute the given points into the general equation $x^2 + y^2 + Dx + Ey + F = 0$ and solve for $D$, $E$, and $F$.For $(-1, 1)$:$$1 + 1 - D + E + F = 0 \Rightarrow -D + E + F = -2$$ (1)For $(-6, -4)$:$$36 + 16 - 6D - 4E + F = 0 \Rightarrow -6D - 4E + F = -52$$ (2)For $(4, -4)$:$$16 + 16 + 4D - 4E + F = 0 \Rightarrow 4D - 4E + F = -32$$ (3)Solving the system of equations (1), (2), and (3) for $D$, $E$, and $F$ gives $D = 2$, $E = 8$, and $F = -8$. Therefore, the equation of the circle is $x^2 + y^2 + 2x + 8y - 8 = 0$.

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 $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$