Question

Find the centre of the smallest circle which cuts orthogonally the two circles x² + y² = 1 and x² + y² + 8x + 8y − 33 = 0. 

Answer 1

Suppose x² + y² + 2gx + 2fy + c = 0 is the required circle. Since this circle cuts orthogonally the given two circles, we have c = 11 and g + f = −4. The radius of the circle is

which is minimum if and only if g = −2. Hence, the centre is (2, 2).