The line joining the points (−1, 4) and (3, 6) is a diameter of the circle C - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 2

The radius r of the circle is half the length of the diameter.

First, we calculate the length of the diameter using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the points (−1, 4) and (3, 6): $d = \sqrt{(3 - (-1))^2 + (6 - 4)^2}$ $= \sqrt{(3 + 1)^2 + (6 - 4)^2}$ $= \sqrt{4^2 + 2^2}$ $= \sqrt{16 + 4}$ $= \sqrt{20} = 2\sqrt{5}$

Thus, the radius is: $r = \frac{d}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5}$

The standard form of the equation of a circle with center (h, k) and radius r is: $(x - h)^2 + (y - k)^2 = r^2$

Substituting the values we found: $h = 1, k = 5, r = \sqrt{5}$

The equation becomes: $(x - 1)^2 + (y - 5)^2 = (\sqrt{5})^2$ $(x - 1)^2 + (y - 5)^2 = 5$

Therefore, the equation of the circle C is: $(x - 1)^2 + (y - 5)^2 = 5$