The line joining the points (-1, 4) and (3, 6) is a diameter of the circle C - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

The radius can be found by measuring the distance from the center (1, 5) to one of the endpoints, say (-1, 4). Using the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Applying the formula:

$$r = \sqrt{(-1 - 1)^2 + (4 - 5)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$$

So, the radius r is (\sqrt{5}).

The standard equation of a circle is given by:

$$(x - h)^2 + (y - k)^2 = r^2$$

where (h, k) is the center and r is the radius. Substituting in our values:

$$(x - 1)^2 + (y - 5)^2 = \left(\sqrt{5}\right)^2$$

This simplifies to:

$$(x - 1)^2 + (y - 5)^2 = 5$$

Thus, the equation of the circle C is:

$$(x - 1)^2 + (y - 5)^2 = 5$$