Question

A line segment through the center of each circle intersects the circle at the points shown. a. Find the coordinates of the circle's center. b. Find the radius of the circle. c. Use your answers from parts (a) and (b) to write the standard form of the circle's equation.

Accepted Answer

Identify the two points on the circle that the line segment passes through. For the first circle, these points are (-5, 9) and (-3, 5). For the second circle, the points are (3, 6) and (5, 4). Find the center of the circle by calculating the midpoint of the line segment connecting the two points. Use the midpoint formula: $$\left( \frac{x_1$ + x_2$}{2}, \frac{y_1$ + y_2$}{2} \right). $$Calculate the radius of the circle by finding the distance from the center to either of the two points on the circle. Use the distance formula: $$\sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$, then divide by 2 since the segment is a diameter. Write the standard form of the circle's equation using the center $$(h, k)$$ and radius $$r$$: $$(x - h)^2 + (y - k)^2 = r^2. $$Substitute the values of the center coordinates and the radius squared into the standard form equation to express the equation of the circle.

A line segment through the center of each circle intersects the circle at the points shown. a. Find the coordinates of the circle's center. b. Find the radius of the circle. c. Use your answers from parts (a) and (b) to write the standard form of the circle's equation.

Verified video answer for a similar problem:

Key Concepts

Midpoint Formula

Distance Formula

Standard Form of a Circle's Equation