Final answer:
The equation of a circle with center (-2,2) and passing through the point (4,-2) is (x + 2)^2 + (y - 2)^2 = 52, which is obtained using the distance formula to find the radius and then substituting into the general equation of a circle.
Step-by-step explanation:
To find the equation of a circle with center (-2,2) and passing through the point (4,-2), you first need to calculate the radius of the circle. The radius (r) can be found by using the distance formula to determine the distance between the center and the point on the circle:
- r = √[(x2 - x1)^2 + (y2 - y1)^2]
- r = √[(4 - (-2))^2 + (-2 - 2)^2]
- r = √[6^2 + (-4)^2]
- r = √[36 + 16]
- r = √52
Now that the radius is known, you can plug it into the general equation of a circle (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle. Substituting the center (-2,2) and radius √52, the equation becomes:
(x + 2)^2 + (y - 2)^2 = 52
This is the desired equation of the circle.