C2 June 2010 Q10
10. The circle \(C\) has centre \(A(2,1)\) and passes through the point \(B(10,\ 7)\).
The line \(l_1\) is the tangent to \(C\) at the point \(B\).
The line \(l_2\) is parallel to \(l_1\) and passes through the mid-point of \(AB\).
Given that \(l_2\) intersects \(C\) at the points \(P\) and \(Q\),
| Scheme | Marks |
|---|---|
| (a) \((10 - 2)^2 + (7 - 1)^2\) or \(\sqrt{(10 - 2)^2 + (7 - 1)^2}\) | M1 A1 |
| \((x \pm 2)^2 + (y \pm 1)^2 = k\) (\(k\) a positive value) | M1 |
| \((x - 2)^2 + (y - 1)^2 = 100\) (Accept \(10^2\) for 100) (Answer only scores full marks) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| (b) (Gradient of radius =) \(\dfrac{7 - 1}{10 - 2} = \dfrac{6}{8}\) (or equiv.) Must be seen in part (b) | B1 |
| Gradient of tangent \(= \dfrac{-4}{3}\) (Using perpendicular gradient method) | M1 |
| \(y - 7 = m(x - 10)\) Eqn., in any form, of a line through (10, 7) with any numerical gradient (except 0 or \(\infty\)) | M1 |
| \(y - 7 = \dfrac{-4}{3}(x - 10)\) or equiv (ft gradient of radius, dep. on both M marks) \(\{3y = -4x + 61\}\) (N.B. The A1 is only available as ft after B0) The unsimplified version scores the A mark (isw if necessary... subsequent mistakes in simplification are not penalised here. The equation must at some stage be exact, not, e.g. \(y = -1.3x + 20.3\) | A1ft |
| (4) |
Notes
(b) 2nd M: Using (10, 7) to find the equation, in any form, of a straight line through (10, 7), with any numerical gradient (except 0 or \(\infty\)).
Alternative: 2nd M: Using (10, 7) and an \(m\) value in \(y = mx + c\) to find a value of \(c\).
(b) Alternative for first 2 marks (differentiation):
\(2(x - 2) + 2(y - 1)\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) or equiv. B1
Substitute \(x\) = 10 and \(y\) = 7 to find a value for \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) M1
(This M mark can be awarded generously, even if the attempted ‘differentiation’ is not ‘implicit’).
| Scheme | Marks |
|---|---|
| (c) \(\sqrt{r^2 - \left(\dfrac{r}{2}\right)^2}\) Condone sign slip if there is evidence of correct use of Pythag. | M1 |
| \(= \sqrt{10^2 - 5^2}\) or numerically exact equivalent | A1 |
| \(PQ\ \left(= 2\sqrt{75}\right) = 10\sqrt{3}\) Simplest surd form \(10\sqrt{3}\) required for final mark | A1 |
| (3) | |
| 11 |
Notes
(c) Alternatives:
To score M1, must be a fully correct method to obtain \(\dfrac{1}{2}PQ\) or \(PQ\).
1st A1: For alternative methods that find \(PQ\) directly, this mark is for an exact numerically correct version of \(PQ\).