C2 June 2005 Q8
8. The circle \(C\), with centre at the point \(A\), has equation \(x^2 + y^2 - 10x + 9 = 0\).
Find
(a) the coordinates of \(A\), (2)
(b) the radius of \(C\), (2)
(c) the coordinates of the points at which \(C\) crosses the \(x\)-axis. (2)
Given that the line \(l\) with gradient \(\tfrac{7}{2}\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),
(d) find an equation of the line which passes through \(A\) and \(T\). (3)
| Scheme | Marks |
|---|---|
| Centre \((5, 0)\) (or \(x = 5,\ y = 0\)) | B1 B1 |
| (2) |
Notes
\((0, 5)\) scores B1 B0.
| Scheme | Marks |
|---|---|
| \((x \pm a)^2 \pm b \pm 9 + (y \pm c)^2 = 0 \Rightarrow r^2 = \ldots\) or \(r = \ldots\), Radius \(= 4\) | M1, A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \((1, 0), \ (9, 0)\) Allow just \(x = 1,\ x = 9\) | B1ft, B1ft |
| (2) |
| Scheme | Marks |
|---|---|
| Gradient of \(AT = -\dfrac{2}{7}\) | B1 |
| \(y = -\dfrac{2}{7}(x - 5)\) | M1 A1ft |
| (3) | |
| (9 marks) |
Notes
M1: Equation of straight line through centre, any gradient (except 0 or \(\infty\)) (The equation can be in any form).
A1ft: Follow through from centre, but gradient must be \(-\dfrac{2}{7}\).