(a) Find the coordinates of the centre and the length of the radius for the circle x2 + y2 − 4x − 8y − 5 = 0.

The equation of a circle is generally given by (x-a)² + (y-b)² = r² where (a,b) is the center of the circle and r is the radius.

The given equation is x² + y² - 4x - 8y - 5 = 0.

To find the center and the radius, we need to rewrite this equation in the form (x-a)² + (y-b)² = r².

First, group the x terms and the y terms:

(x² - 4x) + (y² - 8y) = 5.

To complete the square, we take half the coefficient of x, square it and add it to both sides. We do the same for y:

(x² - 4x + 4) + (y² - 8y + 16) = 5 + 4 + 16.

This simplifies to:

(x-2)² + (y-4)² = 25.

So, the center of the circle (a,b) is (2,4) and the radius r is the square root of 25, which is 5.