June 2022 Paper 3 Q3
3 The points \(P\) and \(Q\) have coordinates \((2, -5)\) and \((3, 1)\) respectively.
Determine the equation of the circle that has \(PQ\) as a diameter. Give your answer in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
| Scheme | Marks | AO |
|---|---|---|
| \(\left(\dfrac{2 + 3}{2}, \dfrac{-5 + 1}{2}\right)\) | M1* | 2.1 |
| \(d^2 = (2 - 3)^2 + (-5 - 1)^2\) | M1* | 1.1 |
| \(\left(x - \dfrac{5}{2}\right)^2 + (y + 2)^2 = \dfrac{37}{4}\) \(x^2 - 5x + \dfrac{25}{4} + y^2 + 4y + 4 = \dfrac{37}{4}\) | M1dep* | 1.1 |
| \(x^2 + y^2 - 5x + 4y + 1 = 0\) | A1 | 2.2a |
| [4] |
Notes
M1*: Attempt at the midpoint (3 out of 4 values used correctly or one correct coordinate)
If correct \(\left(\dfrac{5}{2}, -2\right)\)
M1*: Attempt to calculate diameter \(d\) or radius \(r\) (or the square of either of these two)
e.g. \(r^2 = \left(2 - \frac{5}{2}\right)^2 + (-5 - (-2))^2\)
M0 if implying that \(r^2 = (2 - 3)^2 + (-5 - 1)^2\). If the value is not labelled (as either \(r\) or \(d\)) then consider how this value is used in their circle equation
3 out of 4 values used correctly – if correct \(d^2 = 37\) or \(r^2 = \frac{37}{4}\)
M1dep*: Correct form for the equation of a circle with their values for the centre and radius
Dependent on both previous M marks
A1: Must have \(= 0\)
Order of terms on the lhs irrelevant
\(a\), \(b\) and \(c\) need not be explicitly stated