C2 January 2006 Q3
3.
In Figure 1, \(A(4, 0)\) and \(B(3, 5)\) are the end points of a diameter of the circle \(C\).
Find
(a) the exact length of \(AB\), (2)
(b) the coordinates of the midpoint \(P\) of \(AB\), (2)
(c) an equation for the circle \(C\). (3)
| Scheme | Marks |
|---|---|
| \((AB)^2 = (4 - 3)^2 + (5)^2 \quad [= 26]\) | M1 |
| \(AB = \underline{\sqrt{26}}\) | A1 |
| (2) |
Notes
M1 for an expression for \(AB\) or \(AB^2\) N.B. \((x_1 + x_2)^2 + \ldots\) is M0
| Scheme | Marks |
|---|---|
| \(p = \left(\dfrac{4 + 3}{2}, \dfrac{5}{2}\right)\) | M1 |
| \(= \underline{\left(\dfrac{7}{2}, \dfrac{5}{2}\right)}\) | A1 |
| (2) |
Notes
M1 for a full method for \(x_p\)
| Scheme | Marks |
|---|---|
| \((x - x_p)^2 + (y - y_p)^2 = \left(\dfrac{AB}{2}\right)^2\) LHS | M1 |
| RHS | M1 |
| \((x - 3.5)^2 + (y - 2.5)^2 = 6.5\) oe | A1 c.a.o |
| (3) | |
| (7 marks) |
Notes
1st M1 for using their \(x_p\) and \(y_p\) in LHS
2nd M1 for using their \(AB\) in RHS
N.B. \(x^2 + y^2 - 7x - 5y + 12 = 0\) scores, of course, 3/3 for part (c).
Condone use of calculator approximations that lead to correct answer given.