Final answer:
The standard form equation of the circle with the given diameter endpoints is (x+2)² + (y+4)² = 1.
Step-by-step explanation:
The student is asking for the equation of a circle in standard form, given the endpoints of its diameter. The endpoints provided are (-2,-3) and (-2,-5). First, to find the center of the circle, we take the midpoint of the line segment with the given endpoints; the center (h,k) is then also (-2,-4) because the x-coordinate of both endpoints is the same and the average of the y-coordinates is -4.
The radius of the circle can be found by calculating the distance between the center and one of the endpoints, which is r = √[(x2-x1)² + (y2-y1)²] = √[(-2+2)² + (-4+3)²] = √[0 + (-1)²] = √[1] = 1. The standard form equation of a circle is (x-h)² + (y-k)² = r². Substituting the center (-2,-4) and the radius 1, we get (x+2)² + (y+4)² = 1.