Equation of a circleAQA A-Level Maths: Subtopic test

1. 1
   A circle has equation $(x-3)^2+(y+2)^2=25$.

   (a)

   Write down the centre of the circle.

   [1 mark]

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

   (b)

   Which of these points lies on the circle?

   [1 mark]

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

   (c)

   The line $x=3$ meets the circle at two points. Find their coordinates.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A circle has equation $x^2+y^2-6x+10y+18=0$.

   (a)

   Find the centre of the circle.

   [1 mark]

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

   (b)

   Find the radius of the circle.

   [1 mark]

   • A$16$
   • B$4$
   • C$\sqrt{34}$
   • D$\sqrt{52}$

   (c)

   Show that the circle does not meet the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A circle has centre $C(2,-1)$ and passes through the point $P(5,3)$.

   (a)

   Find the equation of the circle.

   [3 marks]

   (b)

   The circle crosses the $y$-axis at two points. Find the exact $y$-coordinates of these points and the exact distance between them.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A circle $C$ has equation $x^2+y^2-6x+2y-10=0$.

   (a)

   (i) By completing the square, find the centre and exact radius of $C$.
   (ii) Show that the point $A(7,1)$ lies on $C$.
   (iii) $AB$ is a diameter of $C$. Find the coordinates of $B$.

   [6 marks]

   (b)

   The line $y=x-6$ passes through $A(7,1)$ and meets $C$ again at the point $Q$. Find the coordinates of $Q$ and the length of $AQ$.

   [6 marks]

   Total for question 4: 12 marks

End of questions