Find the equation of the circle, written in the form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$ passing through the points (10,3),(5,-2) , and (8,7) .
A) $x ^ { 2 } + y ^ { 2 } - 5 x - 3 y + 9 = 0$
B) $x ^ { 2 } + y ^ { 2 } + 10 x + 6 y + 9 = 0$
C) $x ^ { 2 } + y ^ { 2 } - 10 x - 6 y - 2 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 10 x - 6 y + 9 = 0$

Question

Find the equation of the circle, written in the form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$ passing through the points  (10,3),(5,-2) , and  (8,7) .
A) $x ^ { 2 } + y ^ { 2 } - 5 x - 3 y + 9 = 0$
B) $x ^ { 2 } + y ^ { 2 } + 10 x + 6 y + 9 = 0$
C) $x ^ { 2 } + y ^ { 2 } - 10 x - 6 y - 2 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 10 x - 6 y + 9 = 0$

Quizplus · Accepted Answer

To find the equation of the circle passing through the points (10,3), (5,-2), and (8,7), we can use the general form of the circle equation $x^2 + y^2 + ax + by + c = 0$ and substitute each point into this equation to form a system of equations. Solving this system will give us the values of a, b, and c.Substituting (10,3) into the equation gives: $10^2 + 3^2 + 10a + 3b + c = 0$Substituting (5,-2) into the equation gives: $5^2 + (-2)^2 + 5a - 2b + c = 0$Substituting (8,7) into the equation gives: $8^2 + 7^2 + 8a + 7b + c = 0$Solving this system of equations yields $a = -10$, $b = -6$, and $c = 9$.Therefore, the correct equation of the circle is $x^2 + y^2 - 10x - 6y + 9 = 0$, which matches option D.

Find the Equation of the Circle, Written in the Form x2+y2+ax+by+c=0x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0x2+y2+ax+by+c=0

Find the Equation of the Circle, Written in the Form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$