A-LevelMathematicsCoordinate geometry9709/12 · May/June 2022 · Q88 Marks
The equation of a circle is x² + y² + ax + by − 12 = 0. The points A (1, 1) and B (2, −6) lie on the circle.
- (a)Find the values of a and b and hence find the coordinates of the centre of the circle.[4]
- (b)Find the equation of the tangent to the circle at the point A, giving your answer in the form px + qy = k, where p, q and k are integers.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.
- [B1] 1 + 1 + a + b – 12 = 0 [=> a + b = 10]
- [B1] 4 + 36 + 2a – 6b – 12 = 0 [=> 2a – 6b = -28]
- [B1] a = 4, b = 6
- [B1 FT] Centre is (-'their a'/2, -'their b'/2) [-2, -3]
- [M1] Gradient of AC is (1 - 'their y_c') / (1 - 'their x_c')
- [A1 FT] Gradient of tangent is -1 / 'their m_AC'
- [DM1] Equation: y - 1 = 'their m_tan'(x - 1)
- [A1] 3x + 4y = 7 or 4y + 3x = 7 or integer multiples of these
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