Pure: Coordinate geometry in the (x, y) planeEdexcel A-Level Maths: Topic test

1. 1
   The line $l$ passes through the points $A(-3,5)$ and $B(2,-5)$.

   (a)

   What is the gradient of $l$?

   [1 mark]

   • A$2$
   • B$-2$
   • C$\frac12$
   • D$-\frac12$

   (b)

   Which is the equation of the line through $A$ that is perpendicular to $l$?

   [1 mark]

   • A$2x+y+1=0$
   • B$x+2y-7=0$
   • C$x-2y+13=0$
   • D$2x-y+11=0$

   (c)

   Find an equation of the perpendicular bisector of $AB$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A circle $C$ has equation $x^2+y^2+10x-6y+9=0$.

   (a)

   What are the coordinates of the centre of $C$?

   [1 mark]

   • A$(-5,3)$
   • B$(5,-3)$
   • C$(-10,6)$
   • D$(10,-6)$

   (b)

   What is the radius of $C$?

   [1 mark]

   • A$\sqrt{43}$
   • B$\sqrt{34}$
   • C$25$
   • D$5$

   (c)

   Find the coordinates of the points where $C$ meets the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ has parametric equations $x=4\sin t$, $y=3\cos t-2$, for $0\leqslant t<2\pi$.

   (a)

   Show that a Cartesian equation of $C$ is $\dfrac{x^2}{16}+\dfrac{(y+2)^2}{9}=1$.

   [3 marks]

   (b)

   Find the coordinates of the points where $C$ meets the $y$-axis, and state the range of values of $y$ on $C$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A circle $C$ has parametric equations $x=1+5\cos t$, $y=-2+5\sin t$, for $0\leqslant t<2\pi$.

   (a)

   Find a Cartesian equation of $C$. Hence find the equation of the tangent to $C$ at the point $(4,2)$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [6 marks]

   (b)

   A particle $Q$ moves in a straight line so that, at time $s$ seconds, its position is $x=-4+4s$, $y=-2+2s$, where $s\geqslant0$. Show that $Q$ lies on $C$ at exactly two times, and find the distance between the two positions of $Q$ on $C$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The points $A(1,1)$, $B(7,1)$ and $C(7,9)$ are the vertices of a triangle.

   (a)

   Which side of the triangle is a diameter of its circumcircle?

   [1 mark]

   • A$AC$
   • B$AB$
   • C$BC$
   • DNone of the sides

   (b)

   What is the radius of the circumcircle?

   [1 mark]

   • A$10$
   • B$5$
   • C$\sqrt{10}$
   • D$25$

   (c)

   Find an equation of the circumcircle.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The line $m$ has equation $y=-\frac{2}{3}x+4$.

   (a)

   What is the gradient of a line perpendicular to $m$?

   [1 mark]

   • A$-\frac32$
   • B$\frac23$
   • C$\frac32$
   • D$-\frac23$

   (b)

   At which value of $x$ does $m$ cross the $x$-axis?

   [1 mark]

   • A$4$
   • B$-6$
   • C$\frac83$
   • D$6$

   (c)

   The line $n$ is perpendicular to $m$ and passes through the point $(4,-1)$. Find an equation of $n$ in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A curve $K$ has parametric equations $x=\dfrac{2}{t}$, $y=t+1$, where $t\neq0$.

   (a)

   Find a Cartesian equation of $K$.

   [3 marks]

   (b)

   The line $y=2x-2$ meets $K$ at two points. Find the value of $t$ at each point, and the coordinates of the points.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A restricted zone in the sea has its boundary modelled by the circle $S$ with equation $x^2+y^2-4x+2y-15=0$, where distances are in km.

   (a)

   The points $A(4,3)$ and $B(6,-3)$ lie on the boundary. Find the centre and radius of $S$, and show that the perpendicular bisector of $AB$ passes through the centre of $S$.

   [6 marks]

   (b)

   A boat moves with constant velocity so that, $t$ hours after leaving the point $(0,3)$, its position is $x=2t$, $y=3-2t$. Find the times at which the boat is on the boundary of the zone, the length of time it spends inside the zone, and the speed of the boat.

   [6 marks]

   Total for question 8: 12 marks

End of questions