Find the equation of the circle
$$x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$$
that passes through the points $( 1,9 ) , ( - 4,4 ) , ( 6,4 )$.
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 that passes through the points $(1,9)$, $(-4,4)$, and $(6,4)$, we can substitute these points into the general equation of a circle $x^2 + y^2 + Dx + Ey + F = 0$ and solve for $D$, $E$, and $F$.For $(1,9)$: $1^2 + 9^2 + D(1) + E(9) + F = 0$$$1 + 81 + D + 9E + F = 0$$$$82 + D + 9E + F = 0$$ (1)For $(-4,4)$: $(-4)^2 + 4^2 + D(-4) + E(4) + F = 0$$$16 + 16 - 4D + 4E + F = 0$$$$32 - 4D + 4E + F = 0$$ (2)For $(6,4)$: $6^2 + 4^2 + D(6) + E(4) + F = 0$$$36 + 16 + 6D + 4E + F = 0$$$$52 + 6D + 4E + F = 0$$ (3)Solving these equations simultaneously 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 $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$