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:

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

Substituting the coordinates of A(4, 0) and B(3, 5):

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

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

The general equation of a circle is:

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

where $(h, k)$ is the center and r is the radius. The center of the circle is the midpoint P, and the radius is half the length of AB:

1. Radius calculation: The radius r is given by

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

2. Equation of the circle: Using the coordinates of P for $(h, k)$,

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

This simplifies to:

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

Thus, an equation for the circle C is:

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