Coordinate geometry: past paper questions

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• Question 5May/June 2026, paper 11Structured5 marks

  The equation of a curve is $$y=-\frac{1}{2}x+\sqrt{x^3-3x-2}\mathrm{.}$$ Find an equation of the normal to the curve at the point $(6,11)$.

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• Question 10Oct/Nov 2025, paper 11Structured7 marks

  A circle has equation $x^2+y^2+4y-21=0$ and a straight line has equation $2x+y-8=0$. The line intersects the circle at two points. Find the coordinates of these two points of intersection.

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• Question 2Oct/Nov 2023, paper 11Structured4 marks

  A line has equation $y=2cx+3$ and a curve has equation $y=c{x}^2+3x-c$, where $c$ is a constant. Showing all necessary working, determine which of the following statements is correct.

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• Question 11May/June 2026, paper 11Structured9 marks

  The diagram shows a circle with equation $$(x-8{)}^2+(y-10{)}^2=40.$$ The tangent to the circle at the point $(6,16)$ meets the $y$-axis at the point $P$. The tangent to the circle at the point $(14,8)$ meets the $x$-axis at the point $R$.

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• Question 7Oct/Nov 2025, paper 12Structured11 marks

  The coordinates of the points $P$ and $Q$ are $(1,1)$ and $(7,11)$ respectively. The line segment $PQ$ forms a diameter of a circle. Find the equation of the circle.

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• Question 8Oct/Nov 2024, paper 12Structured10 marks

  The equation of a circle is $x^2+y^2+px+2y+q=0$, where $p$ and $q$ are constants. Express the equation in the form $(x-a{)}^2+(y-b{)}^2=r^2$, where $a$ is to be given in terms of $p$ and $r^2$ is to be given in terms of $p$ and $q$.

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• Question 11Oct/Nov 2023, paper 11Structured9 marksHas a diagram

  The diagram shows the circle with 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 in the form $y=mx+c$.

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• Question 3May/June 2026, paper 12Structured6 marks

  The perpendicular bisector of the line joining $(4,3)$ and $(-2,6)$ meets the line $x-4y-4=0$ at the point $P$. Find the coordinates of $P$.

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• Question 8Oct/Nov 2025, paper 13Structured8 marks

  The points $(6,1)$ and $(-2,7)$ lie at the opposite ends of a diameter of a circle. Find the equation of the circle.

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• Question 11Oct/Nov 2023, paper 12Structured10 marks

  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 value of $p$.

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• Question 8May/June 2026, paper 12Structured11 marks

  A diameter of a circle has end-points at $(-2,4)$ and $(8,8)$. Find an equation of the circle.

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• Question 11Oct/Nov 2025, paper 13Structured13 marks

  The equation of a curve is $y=\frac{8}{3x-8}-\frac{6}{x-1}$. Find the coordinates of the point at which the tangent to the curve at the point $(3,5)$ intersects the line $y=-8x$.

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• Question 10Oct/Nov 2024, paper 13Structured9 marks

  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 $y=\frac{1}{2}x-4$.

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• Question 2Oct/Nov 2023, paper 13Structured4 marks

  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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• Question 1May/June 2026, paper 13Structured5 marks

  Find the coordinates of the points of intersection of the curve $y=3x^2-12x+17$ and the line $y=6x+2$.

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• Question 6May/June 2025, paper 11Structured9 marks

  The equation of a curve is $2x^2-kxy+2=0$ and the equation of a line is $y=px+3$, where $k$ and $p$ are constants. Given that $k=2$ and $p=11$, find the coordinates of the points of intersection of the curve and the line.

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• Question 10May/June 2024, paper 11Structured8 marks

  The equation of a circle is $(x-3{)}^2+y^2=18$. The line with equation $y=mx+c$ passes through the point $(0,-9)$ and is a tangent to the circle.

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• Question 6Oct/Nov 2023, paper 13Structured7 marks

  A line has equation $y=6x-c$ and a curve has equation $y=c{x}^2+2x-3$, where $c$ is a constant. The line is a tangent to the curve at point $P$. Find the possible values of $c$ and the corresponding coordinates of $P$.

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• Question 8May/June 2025, paper 11Structured8 marks

  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 $P$ and $Q$, and the line $x=-2$.

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• Question 11May/June 2024, paper 11Structured13 marksHas a diagram

  A function is defined by $\mathrm{f}(x)=\frac{4}{x^3}-\frac{3}{x}+2$ for $x\mathrm{\ne }0$. The graph of $y=\mathrm{f}(x)$ is shown in the diagram. Find the set of values of $x$ for which $\mathrm{f}(x)$ is decreasing.

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• Question 5May/June 2023, paper 11Structured5 marks

  The line with equation $y=kx-k$, where $k$ is a positive constant, is a tangent to the curve with equation $y=-\frac{1}{2x}$. Find, in either order, the value of $k$ and the coordinates of the point where the tangent meets the curve.

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• Question 11May/June 2026, paper 13Structured12 marksHas a diagram

  The diagram shows a circle with two tangents, $L_1$ and $L_2$. The tangent $L_1$ intersects the $x$-axis at $Q$ and the $y$-axis at $P$. The tangent $L_2$ intersects the $x$-axis at $S$ and the $y$-axis at $R$.

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• Question 8May/June 2025, paper 12Structured9 marksHas a diagram

  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 $C$.

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• Question 7May/June 2024, paper 12Structured8 marks

  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 $a$.

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• Question 10May/June 2023, paper 11Structured11 marksHas a diagram

  The diagram shows part of the curve with equation $y=\frac{4}{(2x-1{)}^2}$ and parts of the lines $x=1$ and $y=1$. The curve passes through the points $A(1,4)$ and $B(\frac{3}{2},1)$.

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• Question 1May/June 2025, paper 13Structured4 marks

  A curve has equation $y=2x+\frac{12}{x^2}$. Find the equation of the tangent to the curve at the point $(-2,-1)$.

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• Question 10May/June 2024, paper 12Structured10 marks

  The equation of a curve is $y=(5-2x{)}^{\frac{3}{2}}+5$ for $x<\frac{5}{2}$. A point $P$ is moving along the curve in such a way that the $y$-coordinate of point $P$ is decreasing at 5 units per second.

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• Question 9May/June 2025, paper 13Structured9 marks

  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 values of $k$ is 10, and find the other possible value.

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• Question 8May/June 2024, paper 13Structured8 marks

  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 $P$.

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• Question 10May/June 2023, paper 12Structured13 marks

  The equation of a circle is $(x-a{)}^2+(y-3{)}^2=20$. The line $y=\frac{1}{2}x+6$ is a tangent to the circle at the point $P$. Show that one possible value of $a$ is 4 and find the other possible value.

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