In the standard (x, y) coordinate plane, a circle passes through the points A(-1, -2) and … — ACT

In the standard $(x, y)$ coordinate plane, a circle passes through the points $A(-1, -2)$ and $B(3, 6)$ . The center of the circle, $C$ , lies on the line with the equation $y = 2x - 5$ . What is the radius of this circle?

Answer: 5

Answer

The radius of the circle is 5.

The perpendicular bisector of the segment connecting

A(-1, -2)

and

B(3, 6)

passes through their midpoint

(1, 2)

with a slope of

-\frac{1}{2}

, giving the equation

x + 2y = 5

. Solving the system of equations with

y = 2x - 5

yields the center at

C(3, 1)

. The distance from

C(3, 1)

A(-1, -2)

\sqrt{(3 - (-1))^2 + (1 - (-2))^2} = \sqrt{4^2 + 3^2} = 5

Step-by-Step Solution

Find the midpoint and slope of the segment

AB

connecting

A(-1, -2)

and

B(3, 6)

Midpoint

M = (1, 2)

and slope

m = 2

The center of any circle passing through

A

and

B

must lie on the perpendicular bisector of segment

AB

Determine the equation of the perpendicular bisector of

AB

x + 2y = 5

The perpendicular bisector passes through the midpoint

M(1, 2)

and has a slope that is the negative reciprocal of the slope of

AB

, which is

-\frac{1}{2}

Find the intersection point of the perpendicular bisector

x + 2y = 5

and the given line

y = 2x - 5

Center

C(3, 1)

The center of the circle lies on both the perpendicular bisector of

AB

and the line

y = 2x - 5

Calculate the distance from the center

C(3, 1)

to point

A(-1, -2)

using the distance formula.

Radius

r = 5

The radius is the distance from the center of the circle to any point on its circumference.

Key Concept

The perpendicular bisector of a chord of a circle passes through the center of that circle.

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