18 The coordinates of two points A and B are (−1, 3) and (5, 7). Find:(a) the coordinates of the midpoint of AB(b) the equation of the circle of which AB is a diameter(c) the coordinates of the intersection points of the circle with the y-axis.

(a) The coordinates of the midpoint of AB can be found by averaging the x-coordinates and the y-coordinates of A and B. The formula for finding the midpoint is ((x1+x2)/2, (y1+y2)/2).

So, the x-coordinate of the midpoint is (-1+5)/2 = 2 and the y-coordinate of the midpoint is (3+7)/2 = 5. Therefore, the coordinates of the midpoint of AB are (2, 5).

(b) The equation of a circle with diameter AB can be found using the formula (x-h)² + (y-k)² = r², where (h, k) are the coordinates of the center of the circle (which is the midpoint of AB) and r is the radius of the circle (which is half the length of AB).

The coordinates of the center of the circle are (2, 5) and the radius can be found using the distance formula: sqrt((x2-x1)² + (y2-y1)²)/2 = sqrt((5-(-1))² + (7-3)²)/2 = sqrt(36+16)/2 = sqrt(52)/2 = sqrt(13).

So, the equation of the circle is (x-2)² + (y-5)² = 13.

(c) The intersection points of the circle with the y-axis are the points where x=0. Substituting x=0 into the equation of the circle gives (0-2)² + (y-5)² = 13, which simplifies to 4 + (y-5)² = 13. Solving for y gives y-5 = ±sqrt(9), so y = 5 ± 3. Therefore, the coordinates of the intersection points of the circle with the y-axis are (0, 2) and (0, 8).