Find the equation of the circle with the points (1, 2) and (5, 6) as the ends of a diameter.

Find the centre and radius of the circle x² + y² − 6x + 4y − 12 = 0.

Find the equation of the circle passing through the origin with its centre at (3, 4).

Find the length of the tangent from the point (5, 6) to the circle x² + y² = 9.

Write the equation of the circle with centre (2, −3) and radius 4.

In a class of 60 students, 35 play cricket, 28 play football and 12 play both. How many play neither?

Prove that cos 2A = 1 − 2sin²A.