Answer:
a. (-3,2)
b. sqrt65
c.
Step-by-step explanation:
a. To find the center of the circle, you can think of it just like finding the midpoint between the two endpoints. To find a midpoint between two endpoints, you take the average of the x values to get the x coordinate, and you take the average of the y values to get the y coordinate of the midpoint. Therefore, if (-10, -2) is (x1, y1) and (4, 6) is (x2, y2), the midpoint/center of the circle would be:
( (x1+x2)/2, (y1+y2)/2 ). When you plug in our x and y values, you get (-3, 2).
b. To find the radius of a circle, you need to know the center/midpoint of the circle which we solved for in part a. The formula for finding the radius of a circle with the center is (x-h)^2 + (y-k)^2 = r^2 for (h, k) as the center. The coordinates of the center that we found earlier for this circle are (-3, 2). With that, we just plug in our numbers into the formula, and we get:
(x+3)^2 + (y-2)^2 = r^2. Now, to get r, we can choose one of the original two endpoints given and plug in the x and y coordinates from that point into this equation. I like (4, 6), so I'm going to plug in 4 for x and 6 for y, and so we get (4+3)^2 + (6-2)^2 = r^2 which equals 49 + 16 = r^2 when simplified. 49 plus 16 is equal to 65, so we get 65 = r^2. To finally get r, we square root both sides of the equation to get r = sqrt65 which is already in the simplest radical form.
c. The circle equation is (x-h)^2 + (y-k)^2 = r^2, like I said in part b. Therefore, we already have our circle equation! We just plug in our center points and we get (x+3)^2 + (y-2)^2 = sqrt65. This is usually an equation a question will give you for a circle, and with this information, they will expect you to find the center (h, k) or the circle and it's radius, r.
