P2: Coordinate geometry in the (x, y) planeEdexcel International A Level Maths: Topic test

1. 1
   A circle $C$ has equation $x^2+y^2+6x-4y-12=0$.

   (a)

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

   [1 mark]

   • A$(-3,2)$
   • B$(3,-2)$
   • C$(6,-4)$
   • D$(-6,4)$

   (b)

   What is the radius of $C$?

   [1 mark]

   • A$25$
   • B$1$
   • C$5$
   • D$2\sqrt3$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points $P(-2,-3)$ and $Q(6,3)$ are the ends of a diameter of a circle $S$. The point $R(5,4)$ lies on $S$.

   (a)

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

   [1 mark]

   • A$(4,0)$
   • B$(2,0)$
   • C$(2,3)$
   • D$(8,6)$

   (b)

   What is the size of angle $PRQ$?

   [1 mark]

   • A$45^\circ$
   • B$60^\circ$
   • C$180^\circ$
   • D$90^\circ$

   (c)

   Find an equation of $S$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A circle $C$ has centre $(3,-1)$ and passes through the point $A(7,2)$.

   (a)

   Find an equation of $C$, and show that the point $(-1,2)$ lies on $C$.

   [3 marks]

   (b)

   Find an equation of the tangent to $C$ at $A$, giving your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A circle $C$ has equation $x^2+y^2-8x+2y-8=0$. The line $l$ with equation $y=x+2$ meets $C$ at the points $A$ and $B$.

   (a)

   Find the coordinates of the centre and the radius of $C$, and the coordinates of $A$ and $B$.

   [6 marks]

   (b)

   The points $A(0,2)$ and $B(1,3)$ are the points where $l$ meets $C$. Show that the perpendicular from the centre of $C$ to the chord $AB$ meets $AB$ at its midpoint, and find the exact distance from the centre of $C$ to $AB$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A circle has centre $(-2,5)$ and radius $\sqrt{13}$.

   (a)

   Which of the following is an equation of the circle?

   [1 mark]

   • A$(x-2)^2+(y+5)^2=13$
   • B$(x+2)^2+(y-5)^2=\sqrt{13}$
   • C$(x+2)^2+(y-5)^2=169$
   • D$(x+2)^2+(y-5)^2=13$

   (b)

   Which of the following points lies on the circle?

   [1 mark]

   • A$(1,7)$
   • B$(1,5)$
   • C$(-2,8)$
   • D$(0,3)$

   (c)

   Show that the equation of the circle can be written as $x^2+y^2+4x-10y+16=0$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A circle $T$ has centre $(2,-1)$ and radius $\sqrt{20}$. The point $A(4,3)$ lies on $T$.

   (a)

   What is the gradient of the tangent to $T$ at $A$?

   [1 mark]

   • A$2$
   • B$\dfrac12$
   • C$-\dfrac12$
   • D$-2$

   (b)

   Which of the following is an equation of the tangent to $T$ at $A$?

   [1 mark]

   • A$2x-y=5$
   • B$x+2y=10$
   • C$x-2y=-2$
   • D$2x+y=11$

   (c)

   Find the coordinates of the other end of the diameter of $T$ that passes through $A$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A circle $D$ has equation $(x-3)^2+(y+2)^2=17$. The points $E(7,-1)$ and $F(-1,-3)$ are given.

   (a)

   Show that $E$ and $F$ are the ends of a diameter of $D$.

   [3 marks]

   (b)

   The point $G(4,2)$ lies on $D$. Show that angle $EGF$ is a right angle, and state the circle property this illustrates.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The points $A(0,1)$, $B(0,7)$ and $D(6,7)$ lie on a circle $C$.

   (a)

   Show that $AD$ is a diameter of $C$, and find an equation of $C$.

   [6 marks]

   (b)

   The tangent to $C$ at $D$ meets the $y$-axis at $T$. Find the area of triangle $ADT$.

   [6 marks]

   Total for question 8: 12 marks

End of questions