In Figure 1, A(4, 0) and B(3, 5) are the end points of a diameter of the circle C - Edexcel - A-Level Maths Pure - Question 5 - 2006 - Paper 2

The coordinates of the midpoint P can be calculated using the midpoint formula:

$P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

For points A(4, 0) and B(3, 5):

$P = \left( \frac{4 + 3}{2}, \frac{0 + 5}{2} \right) = \left( \frac{7}{2}, \frac{5}{2} \right)$

Thus, the coordinates of the midpoint P are $\left( \frac{7}{2}, \frac{5}{2} \right)$.

The general equation of a circle with center $(h, k)$ and radius $r$ is given by:

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

Here, P($\frac{7}{2}, \frac{5}{2}$) is the midpoint which is the center of the circle, and the radius is half the length of AB:

$r = \frac{AB}{2} = \frac{\sqrt{26}}{2}$

Substituting the values into the circle equation:

$\left(x - \frac{7}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = \left(\frac{\sqrt{26}}{2}\right)^2$

Simplifying further gives:

$\left(x - \frac{7}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = \frac{26}{4} = 6.5$

Thus, the equation for the circle C is: $\left(x - \frac{7}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = 6.5$