Using the fact that the centre of a circle is the midpoint of a diameter, find the equation of the circle with thediameter endpoints given.(a) (3, 4) and (9,−6)

Sure, here are the steps to find the equation of the circle:

Step 1: Find the midpoint of the diameter. The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). For the points (3, 4) and (9, -6), the midpoint would be ((3+9)/2, (4-6)/2) = (6, -1). So, the center of the circle is at (6, -1).

Step 2: Find the radius of the circle. The radius is the distance from the center of the circle to either endpoint of the diameter. You can use the distance formula for this: sqrt((x2 - x1)^2 + (y2 - y1)^2). For the points (6, -1) and (3, 4), the radius would be sqrt((3-6)^2 + (4 - (-1))^2) = sqrt(9 + 25) = sqrt(34).

Step 3: Write the equation of the circle. The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. Substituting the values we found, the equation of the circle is (x - 6)^2 + (y + 1)^2 = 34.