June 2024 Paper 2 Q12
12 The equation of a circle is \(\quad (x - 2)^2 + (y + 3)^2 = 16\)
The equation of a line is \(\quad y = 4 - x\)
The circle and line intersect at two points, \(A\) and \(B\).
(a) Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(\quad 2x^2 - 18x + 37 = 0\) [3 marks]
(b)
For \(A\), the \(x\)-coordinate and \(y\)-coordinate are both positive.
Work out the coordinates of \(A\).
Give each coordinate to 2 decimal places. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 - 4x + 4\) or \(y^2 + 6y + 9\) or \(49 - 14x + x^2\) | M1 | oe expansion eg \(x^2 - 2x - 2x + 4\) or \(y^2 + 3y + 3y + 9\) or \(49 - 7x - 7x + x^2\) may be embedded eg in an equation |
| \(x^2 - 4x + 4 + 49 - 14x + x^2 = 16\) or \(x^2 - 4x + 4 + 16 - 8x + x^2 + 24 - 6x + 9 = 16\) | M1dep | oe equation in \(x\) with all relevant expansions seen eg \(x^2 - 2x - 2x + 4 + 49 - 7x - 7x + x^2 = 16\) |
| M2 seen and \(2x^2 - 18x + 37 = 0\) with no rearrangement errors seen | A1 |
Additional guidance
| After M1 an expression may be simplified when forming an equation eg \(x^2 - 4x + 4 + 49 - 14x + x^2\) \(2x^2 - 18x + 53 = 16\) (equation with all relevant expansions seen) \(2x^2 - 18x + 37 = 0\) (no rearrangement errors seen) | M1 M1 A1 |
| For M1dep the oe equation cannot be \(2x^2 - 18x + 37 = 0\) |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{--18 \pm \sqrt{(-18)^2 - 4 \times 2 \times 37}}{2 \times 2}\) or \(\dfrac{18 \pm \sqrt{28}}{4}\) or [5.8, 5.823] or [3.17, 3.2] | M1 | oe calculation or value for at least one \(x\)-coordinate eg \(\dfrac{9 \pm \sqrt{7}}{2}\) or \(\dfrac{9}{2} \pm \sqrt{\dfrac{7}{4}}\) condone \(+\) for \(\pm\) condone \(-\) for \(\pm\) |
| \(4 - \dfrac{18 + \sqrt{28}}{4}\) or \(4 -\) their [5.8, 5.823] or [−1.823, −1.8] or \(4 - \dfrac{18 - \sqrt{28}}{4}\) or \(4 -\) their [3.17, 3.2] or [0.8, 0.83] | M1dep | oe calculation or value for at least one \(y\)-coordinate |
| (3.18, 0.82) | A1 | both coordinates must be to 2 dp |
Additional guidance
| Marks may be awarded from working in (a) | |
| Using \(-18^2\) in the discriminant is M0 unless recovered | |
| Substitution into equation of circle is possible for the 2nd M1 eg \((y + 3)^2 = 16 - (3.17 - 2)^2\) (1st M1 for 3.17 but no calculation or value of \(y\)) \((y =)\ \sqrt{16 - (3.17 - 2)^2} - 3\) | M1 M1 |