To find the equation of a circle given two points A and B that are on the diameter, we will apply the distance formula to find the diameter, the midpoint formula for the center of the circle, and finally use the standard equation of a circle.
Given points:
- A(-8, -8)
- B(-2, -8)
Step 1: Find the distance (diameter)
Using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting coordinates of points A and B:
- \(x_1 = -8\)
- \(y_1 = -8\)
- \(x_2 = -2\)
- \(y_2 = -8\)
Calculating: \[ d = \sqrt{((-2) - (-8))^2 + ((-8) - (-8))^2} \] \[ = \sqrt{(-2 + 8)^2 + (0)^2} \] \[ = \sqrt{(6)^2 + 0} \] \[ = \sqrt{36} \] \[ = 6 \]
Thus, the measure of the diameter of the circle is 6.
Step 2: Calculate the radius
The radius \(r\) is half the diameter: \[ r = \frac{d}{2} = \frac{6}{2} = 3 \]
Thus, the measure of the radius of the circle is 3.
Step 3: Find the center of the circle (h, k)
Using the midpoint formula: \[ (h, k) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of points A and B: \[ (h, k) = \left(\frac{-8 + (-2)}{2}, \frac{-8 + (-8)}{2}\right) \] \[ = \left(\frac{-10}{2}, \frac{-16}{2}\right) \] \[ = (-5, -8) \]
Thus, the center of the circle is \((-5, -8)\).
Step 4: Write the entire equation of the circle
Using the standard equation of a circle: \[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = -5\), \(k = -8\), and \(r = 3\): \[ (x - (-5))^2 + (y - (-8))^2 = 3^2 \] which simplifies to: \[ (x + 5)^2 + (y + 8)^2 = 9 \]
Final Outputs
- Measure of the diameter of the circle: 6
- Measure of the radius of the circle: 3
- Center of the circle (h, k): (-5, -8)
- Equation of the circle: \((x + 5)^2 + (y + 8)^2 = 9\)