Coordinate geometry — AS & A Level Mathematics (9709) questions by topic
Finding the distance between the centres of two circles given in general or completed-square form, locating a second circle's centre where a tangent meets an axis, and using a parallelogram's given gradient to find an unknown coordinate are typical of the 20 coordinate geometry questions from 2023 to 2025, worth 4 to 13 marks.
Find, show, write and express are the command words tracked here, each requiring an equation manipulated into a specific form rather than just a numerical answer. Because converting a circle's general equation into centre-radius form is a frequent early step, submit your working for instant marking, so a wrongly completed square or a misidentified centre is caught before it derails a longer geometry question built on it.
May–June 2025, Paper 11
The circle with equation x^2 + y^2 − 6x + 10y − 27 = 0 intersects the line x = −2 at the points P and Q. Find the area of the triangle formed by the tangents to the circle at P…
Answer this question and get it marked →May–June 2025, Paper 12
Find the coordinates of the points of intersection of the curve and the line with equations 2xy + 5y^2 = 24 and 2x + y + 4 = 0.
Answer this question and get it marked →The diagram shows the circle with equation x^2 + y^2 − 14x + 8y + 36 = 0 and the line y = −2. The line intersects the circle at the points A and B. The centre of the circle is C.…
Answer this question and get it marked →May–June 2025, Paper 13
Three points P, Q and R have coordinates P(−13, 5), Q(5, 1) and R(2, k), where k is a constant. It is given that the angle PRQ is a right angle. Show that one of the possible…
Answer this question and get it marked →May–June 2025, Paper 15
In the parallelogram ABCD, the coordinates of A are (3, 7), the coordinates of B are (6, p) and the coordinates of D are (1, p). It is given that the gradient of AB is −2/3. Find…
Answer this question and get it marked →The equation of a circle is x^2 + y^2 + 4x − 8y − 12 = 0. Find an equation of the tangent to the circle at the point (2, 8), giving your answer in the form ax + by + c = 0. [4]
Answer this question and get it marked →May–June 2024, Paper 12
The equation of a circle is (x − 6)² + (y + a)² = 18. The line with equation y = 2a − x is a tangent to the circle. Find the two possible values of the constant a.
Answer this question and get it marked →May–June 2024, Paper 13
A circle with equation x^2 + y^2 − 6x + 2y − 15 = 0 meets the y-axis at the points A and B. The tangents to the circle at A and B meet at the point P. Find the coordinates of P.
Answer this question and get it marked →October–November 2024, Paper 11
Circles C₁ and C₂ have equations x² + y² + 6x − 10y + 18 = 0 and (x − 9)² + (y + 4)² − 64 = 0 respectively. Find the distance between the centres of the circles.
Answer this question and get it marked →October–November 2024, Paper 12
The equation of a circle is x² + y² + px + 2y + q = 0, where p and q are constants. Express the equation in the form (x − a)² + (y − b)² = r², where a is to be given in terms of p…
Answer this question and get it marked →October–November 2024, Paper 13
The diagram shows the curves with equations y = x³ − 3x + 3 and y = 2x³ − 4x² + 3. Find the x-coordinates of the points of intersection of the curves.
Answer this question and get it marked →Points A and B have coordinates (4, 3) and (8, −5) respectively. A circle with radius 10 passes through the points A and B. Show that the centre of the circle lies on the line y =…
Answer this question and get it marked →May–June 2023, Paper 11
The line with equation y = kx − k, where k is a positive constant, is a tangent to the curve with equation y = −1/(2x). Find, in either order, the value of k and the coordinates…
Answer this question and get it marked →The diagram shows a circle P with centre (0, 2) and radius 10 and the tangent to the circle at the point A(6, 10). A second circle Q has centre at the point where this tangent…
Answer this question and get it marked →May–June 2023, Paper 12
The equation of a circle is (x − a)² + (y − 3)² = 20. The line y = (1/2)x + 6 is a tangent to the circle at the point P. Show that one possible value of a is 4 and find the other…
Answer this question and get it marked →May–June 2023, Paper 13
A circle has equation (x − 1)² + (y + 4)² = 40. A line with equation y = x − 9 intersects the circle at points A and B. Find the coordinates of the two points of intersection.
Answer this question and get it marked →October–November 2023, Paper 11
The circle has equation (x − 4)² + (y + 1)² = 40. Parallel tangents, each with gradient 1, touch the circle at points A and B. Find the equation of the line AB, giving the answer…
Answer this question and get it marked →October–November 2023, Paper 12
The diagram shows curves with equations y = 2x^(1/2) + 13x^(−1/2) and y = 3x^(−1/2) + 12. The curves intersect at points A and B. Find the coordinates of A and B.
Answer this question and get it marked →The coordinates of points A, B and C are (6, 4), (p, 7) and (14, 18) respectively, where p is a constant. The line AB is perpendicular to the line BC. Given that p < 10, find the…
Answer this question and get it marked →October–November 2023, Paper 13
The circle with equation (x − 3)² + (y − 5)² = 40 intersects the y-axis at points A and B. Find the y-coordinates of A and B, expressing your answers in terms of surds.
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