Question

Two circles have their centers at (2, 4) and (-14, 2) and they intersect at the point (-2, 7). What is the radius of each circle?

Answer 1

To determine the radius of each circle, find the distance of the intersection and the center of the circles.

Thus, the radius of the circle with center (2,4) is as follows:

`\begin{gathered} r_1=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ =\sqrt[]{(-2-2)^2+(7-4)^2} \\ =\sqrt[]{(-4)^2+(3)^2} \\ =\sqrt[]{16+9} \\ =\sqrt[]{25} \\ =5 \end{gathered}`

Thus, the radius of the circle with center (-14,2) is as follows:

`\begin{gathered} r_2=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ =\sqrt[]{\lbrack-2-(-14)\rbrack^2+(7-2)^2} \\ =\sqrt[]{(-2+14)^2+(7-2)^2} \\ =\sqrt[]{(12)^2+(5)^2} \\ =\sqrt[]{144+25} \\ =\sqrt[]{169} \\ =13 \end{gathered}`

Thus, the radius of the circles with centers at (2,4) and (-14,2) passing through (-2,7) are 5 and 13, respectively.