Final answer:
The equation of the smallest circle passing through points A and B, which are the intersection points of the given circle x²+y²−4x−8y=0. Therefore, the correct equation is (c) x²+y²=5.
Step-by-step explanation:
The question asks for the equation of the smallest circle passing through points A and B, where A and B are intersection points of circles x²+y²−4x−8y=0 and another given circle. To solve this, we need to find the equation of a circle that passes through the intersection points of the first circle with each of the options provided.
The smallest circle would have the smallest radius among the options that actually intersect with the first circle. Without calculating the specific intersection points and the center of the circle, we can compare the distances of the centers of the given circles to their radii.
The first circle's center is at (2, 4) and it has a radius of 4√2. Looking at the options, the one with a radius smaller than 4√2 but still capable of intersecting the first circle would be the correct answer. Therefore, the correct equation is (c) x²+y²=5, which represents a circle with a radius of √5.