June 2023 Paper 2 Q6
6 The parametric equations of a circle are
\(x = 2\cos\theta - 3\) and \(y = 2\sin\theta + 1\).
Determine the cartesian equation of the circle in the form \((x-a)^2 + (y-b)^2 = k\), where \(a\), \(b\) and \(k\) are integers. [4]
| Scheme | Marks | AO |
|---|---|---|
| \(2\cos\theta = x + 3\) or \(\cos\theta = \frac{x+3}{2}\) | B1 | 2.1 |
| \(2\sin\theta = y - 1\) or \(\sin\theta = \frac{y-1}{2}\) | B1 | 1.1 |
| \(\left(\frac{x \pm 3}{2}\right)^2 + \left(\frac{y \pm 1}{2}\right)^2 = \cos^2\theta + \sin^2\theta\) or \((x \pm 3)^2 + (y \pm 1)^2 = 4\cos^2\theta + 4\sin^2\theta\) oe | M1 | 1.1 |
| \((x+3)^2 + (y-1)^2 = 4\) | A1 | 1.1 |
| [4] |
Notes
M1: allow sign errors in their expressions for \(\sin\theta\) and \(\cos\theta\); allow if just see brackets expanded, but must be 3 terms in each case
A1: allow SC2 for \((x+3)^2 + (y-1)^2 = 4\) unsupported
Alternatively
| Scheme | Marks |
|---|---|
| centre of circle is \((-3, 1)\) | B1 |
| radius is 2 | B1 |
| \((x+3)^2 + (y-1)^2 = 2^2\) | M1 |
| \((x+3)^2 + (y-1)^2 = 4\) | A1 |
M1: allow one sign error in bracket;