June 2025 Paper 1 Q22
22 A circle has centre \((2, 3)\) and radius 5
A straight line passes through \((3, 7)\) and has a gradient of 2
Work out the coordinates of the two points where the circle and the straight line intersect. [6 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(y = 2x + 1\) | B1 | oe equation of straight line eg \(y - 7 = 2(x - 3)\) |
| \((x - 2)^2 + (y - 3)^2 = 5^2\) | B1 | oe eg \(= 25\) or brackets expanded correctly |
| Substitutes their \((2x + 1)\) into their equation of a circle in the form \((x - a)^2 + (y - b)^2 =\) their \(5^2\) where \(a\) and \(b \ne 0\) | M1 | eg \((x - 2)^2 + ((2x + 1) - 3)^2 = 5^2\) or \(x^2 - 4x + 4 + (2x + 1)^2 - 6(2x + 1) + 9 = 25\) |
| \(5x^2 - 12x - 17\ (= 0)\) | A1 | condone \(5x^2 - 12x = 17\) if completing the square |
| \((5x - 17)(x + 1)\ (= 0)\) or \(\dfrac{12 \pm \sqrt{144 + 340}}{10}\) or \(5\left[\left(x - \dfrac{6}{5}\right)^2 - \dfrac{121}{25}\right]\) | M1dep | dep on B2M1A1 |
| \((-1, -1)\) and \((3.4, 7.8)\) | A1 | oe |
| Alternative method 2 | ||
| \(x = \dfrac{y - 1}{2}\) | B1 | oe equation of straight line eg \(y - 7 = 2(x - 3)\) |
| \((x - 2)^2 + (y - 3)^2 = 5^2\) | B1 | oe eg \(= 25\) or brackets expanded correctly |
| Substitutes their \(\dfrac{y - 1}{2}\) into their equation of a circle in the form \((x - a)^2 + (y - b)^2 =\) their \(5^2\) where \(a\) and \(b \ne 0\) | M1 | eg \(\left(\left(\dfrac{y - 1}{2}\right) - 2\right)^2 + (y - 3)^2 = 5^2\) or \(\left(\dfrac{y - 1}{2}\right)^2 - 4\left(\dfrac{y - 1}{2}\right) + 4 + y^2 - 6y + 9 = 25\) |
| \(5y^2 - 34y - 39\ (= 0)\) | A1 | condone \(5y^2 - 34y = 39\) if completing the square |
| \((5y - 39)(y + 1)\ (= 0)\) or \(\dfrac{34 \pm \sqrt{1156 + 780}}{10}\) or \(5\left[\left(y - \dfrac{17}{5}\right)^2 - \dfrac{484}{25}\right]\) | M1dep | dep on B2M1A1 would imply B2M1A1 |
| \((-1, -1)\) and \((3.4, 7.8)\) or \(\left(\dfrac{17}{5}, \dfrac{39}{5}\right)\) | A1 | |
Additional guidance
| First M mark can imply the B marks if correct | |
| Expanding the circle equation and then substituting in can gain marks as an oe but won’t if the circle equation has been expanded incorrectly | |
| Condone \((5x - 17)(5x + 5)\ (= 0)\) and \((5y - 39)(5y + 5)\ (= 0)\) for the fifth mark | |
| Correct answer from Trial and Improvement | B2M2A2 |