The endpoints of a diameter of a circle are \( (-10,-10) \) and \( (2,6) \) (a) Write an equation of the circle in standard form. (b) Graph the circle. Part: \( 0 / 2 \)

To find the equation of the circle given the endpoints of its diameter, we can follow these steps: ### (a) Write an equation of the circle in standard form. 1. **Find the center of the circle**: The center of the circle is the midpoint of the diameter. The midpoint \( M \) of the endpoints \( (-10, -10) \) and \( (2, 6) \) can be calculated using the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates: \[ M = \left( \frac{-10 + 2}{2}, \frac{-10 + 6}{2} \right) = \left( \frac{-8}{2}, \frac{-4}{2} \right) = (-4, -2) \] So, the center of the circle is \( (-4, -2) \). 2. **Find the radius of the circle**: The radius is half the distance between the endpoints of the diameter. We can find the distance \( d \) between the two points using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates: \[ d = \sqrt{(2 - (-10))^2 + (6 - (-10))^2} = \sqrt{(2 + 10)^2 + (6 + 10)^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \] The radius \( r \) is half of this distance: \[ r = \frac{d}{2} = \frac{20}{2} = 10 \] 3. **Write the equation of the circle**: The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting \( h = -4 \), \( k = -2 \), and \( r = 10 \): \[ (x + 4)^2 + (y + 2)^2 = 10^2 \] Thus, the equation of the circle is: \[ (x + 4)^2 + (y + 2)^2 = 100 \] ### (b) Graph the circle. To graph the circle, follow these steps: 1. **Plot the center**: The center of the circle is at the point \( (-4, -2) \). 2. **Draw the radius**: From the center, measure 10 units in all directions (up, down, left, right) to find points on the circle: - Up: \( (-4, -2 + 10) = (-4, 8) \) - Down: \( (-4, -2 - 10) = (-4, -12) \) - Left: \( (-4 - 10, -2) = (-14, -2) \) - Right: \( (-4 + 10, -2) = (6, -2) \) 3. **Sketch the circle**: Draw a smooth curve connecting these points, ensuring the circle is centered at \( (-4, -2) \) and has a radius of 10. This completes the problem. The equation of the circle is: \[ \boxed{(x + 4)^2 + (y + 2)^2 = 100} \]