October 2021 Paper 2 Q7
7 The parametric equations of a circle are
\(x = 7 + 5\cos\theta\), \(\quad y = 5\sin\theta - 3\), \(\quad\) for \(0 \leqslant \theta \leqslant 2\pi\).
(a) Find a cartesian equation of the circle. [3]
(b) State the coordinates of the centre of the circle. [1]
| Scheme | Marks | AO |
|---|---|---|
| \((x-7)^2 + (y+3)^2 = 5^2\cos^2\theta + 5^2\sin^2\theta\) oe | M1 | 3.1a |
| use of \(\cos^2\theta + \sin^2\theta = 1\) to eliminate \(\theta\) | M1 | 1.1 |
| \((x-7)^2 + (y+3)^2 = 5^2\) oe isw | A1 | 1.1 |
| [3] |
Notes
M1: allow sign error
Alternative
| Scheme | Marks |
|---|---|
| centre is (7, –3) and substituted in correct form of equation | M1 |
| radius is 5 and substituted in correct form of equation | M1 |
A1: if M0M0 allow SC1 for
\(y = 5\sin\left\{\cos^{-1}\left(\dfrac{x-7}{5}\right)\right\} - 3\)
\(x = 5\cos\left\{\sin^{-1}\left(\dfrac{y+3}{5}\right)\right\} + 7\)
| Scheme | Marks | AO |
|---|---|---|
| (7, –3) | B1 | 2.2a |
| [1] |
Notes
B1: FT their \((x-7)^2 + (y+3)^2 = 25\)