Find the center and the radius of the circle that passes through the points $( - 4,12 ) , ( 10 , - 2 ) \text { and } ( 2 , - 6 )$
A) $\text { The center is at } ( - 3,2 ) \text { and the radius is } 2 .$
B) The center is at $( 0,2 )$ and the radius is $0 .$
C) The center is at $( 3,3 )$ and the radius is $1 .$
D) The center is at $( 0,0 )$ and the radius is 7.
E) The center is at $( 2,4 )$ and the radius is $10 .$

Question

Quizplus · Accepted Answer

We can use the fact that the center of a circle is the intersection of its perpendicular bisectors. Let's find the equation of the perpendicular bisectors of the segment joining the points $(-4,12)$ and $(10,-2)$ and the segment joining the points $(10,-2)$ and $(2,-6)$.

For the first segment, we find that the midpoint is $(3,5)$ and the slope is $-\frac{14}{7}=-2$. Therefore, the equation of the perpendicular bisector is of the form $y-5=-(1/2)(x-3)$, or $x+2y=11$.

For the second segment, we find that the midpoint is $(6,-4)$ and the slope is $\frac{4}{8}=\frac{1}{2}$. Therefore, the equation of the perpendicular bisector is of the form $y+4=(1/2)(x-6)$, or $x-2y=18$.

The intersection of these two lines is $(2,4)$, which is the center of the circle. Finally, we compute the distance between $(2,4)$ and one of the given points, say $(10,-2)$, to find the radius. We get $\sqrt{(10-2)^2+(-2-4)^2}=10$, so the answer is (E).

Find the Center and the Radius of the Circle That (−4,12),(10,−2) and (2,−6)( - 4,12 ) , ( 10 , - 2 ) \text { and } ( 2 , - 6 )(−4,12),(10,−2) and (2,−6)

Find the Center and the Radius of the Circle That $( - 4,12 ) , ( 10 , - 2 ) \text { and } ( 2 , - 6 )$