Test On The Equation Of Circle

The equation of a circle with center (h, k) and radius r is given by (x-h)^2 + (y-k)^2 = r^2. In this case, the center is (-1, 2) and the radius is 2 units. Plugging these values into the equation, we get (x+1)^2 + (y-2)^2 = 2^2. Simplifying further, we have (x+1)^2 + (y-2)^2 = 4. Therefore, the correct options are Option 1 and Option 3, as they represent the correct equation of the circle.

The equation of a circle can be found using the formula (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. Since the line segment AB is the diameter of the circle, the midpoint of AB will be the center of the circle. The midpoint of AB can be found using the midpoint formula, which is ((x1 + x2)/2, (y1 + y2)/2). Plugging in the values of A and B, we get the midpoint as ((-2 + 4)/2, (0 + 8)/2) = (1, 4). Therefore, the center of the circle is (1, 4). To find the radius, we can use the distance formula between A and B, which is sqrt((x2 - x1)^2 + (y2 - y1)^2). Plugging in the values of A and B, we get the radius as sqrt((4 - (-2))^2 + (8 - 0)^2) = sqrt(36 + 64) = sqrt(100) = 10. Therefore, the equation of the circle is (x - 1)^2 + (y - 4)^2 = 10^2, which simplifies to (x - 1)^2 + (y - 4)^2 = 100. This corresponds to Option 1 and Option 3.