A-LevelMathematicsCoordinate geometry9709/12 · May/June 2020 · Q1111 Marks
The equation of a circle with centre C is x² + y² – 8x + 4y − 5 = 0.
- (a)Find the radius of the circle and the coordinates of C.[3]
- (b)The point P (1, 2) lies on the circle. Show that the equation of the tangent to the circle at P is 4y = 3x + 5.[3]
- (c)The point Q also lies on the circle and PQ is parallel to the x-axis. Write down the coordinates of Q.[2]
- (d)The tangents to the circle at P and Q meet at T. Find the coordinates of T.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.
- [M1] Express as (x - 4)² + (y + 2)² = 16 + 4 + 5
- [A1] Centre C(4, -2)
- [A1] Radius = √25 = 5
- [B1FT] P(1,2) to C(4, -2) has gradient = -4/3 (FT on coordinates of C)
- [M1] Tangent at P has gradient = 3/4
- [A1] Equation is y - 2 = (3/4)(x - 1) or 4y = 3x + 5
- [B1] Q has the same coordinate as P y = 2
- [B1] Q is as far to the right of C as P x = 3 + 3 + 1 = 7 Q (7, 2)
- [B1FT] Gradient of tangent at Q = -3/4 by symmetry (FT from part (b))
- [M1] Eqn of tangent at Q is y - 2 = (-3/4)(x - 7) or 4y + 3x = 29
- [A1] T(4, 17/4)
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