C2 June 2006 Q7
7.
The line \(y = 3x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(P(2, 2)\), as shown in Figure 1.
The point \(Q\) is the centre of \(C\).
Given that \(Q\) lies on the line \(y = 1\),
| Scheme | Marks |
|---|---|
| Gradient of \(PQ\) is \(-\dfrac{1}{3}\) | B1 |
| \(y - 2 = -\dfrac{1}{3}(x - 2) \qquad (3y + x = 8)\) | M1 A1 |
| (3) |
Notes
M1: eqn. of a straight line through \((2, 2)\) with any gradient except 3, 0 or \(\infty\).
Alternative: Using \((2, 2)\) in \(y = mx + c\) to find a value of \(c\) scores M1, but an equation (general or specific) must be seen.
If the given value \(x = 5\) is used to find the gradient of \(PQ\), maximum marks are (a) B0 M1 A1 (b) B0.
| Scheme | Marks |
|---|---|
| \(y = 1: \quad 3 + x = 8 \quad x = 5\) (*) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \((\text{"}5\text{"} - 2)^2 + (1 - 2)^2\) M: Attempt \(PQ^2\) or \(PQ\) | M1 A1 |
| \((x - 5)^2 + (y - 1)^2 = 10\) M: \((x \pm a)^2 + (y \pm b)^2 = k\) | M1 A1 |
| (4) | |
| (8 marks) |
Notes
For the first M1, condone one slip, numerical or sign, inside a bracket.
The first M1 can be scored if their \(x\)-coord. is used instead of 5.
For the second M1, allow any equation in this form, with non-zero \(a\), \(b\) and \(k\).