Circle propertiesEdexcel International A Level Maths: Subtopic test

1. 1
   A circle has diameter $AB$, where $A=(1,2)$ and $B=(9,8)$. The point $P=(8,1)$ lies on the circle.

   (a)

   Find the gradient of $AP$.

   [1 mark]

   • A$\frac17$
   • B$7$
   • C$-\frac17$
   • D$-7$

   (b)

   Which of the following is an equation of the circle?

   [1 mark]

   • A$(x-5)^2+(y-5)^2=25$
   • B$(x-5)^2+(y-5)^2=5$
   • C$(x+5)^2+(y+5)^2=25$
   • D$(x-5)^2+(y-5)^2=100$

   (c)

   Show that angle $APB$ is a right angle, and name the circle property you are using.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A circle has centre $C=(2,3)$ and radius $5$. The points $P=(5,7)$ and $Q=(6,0)$ lie on the circle.

   (a)

   Find the coordinates of the midpoint $M$ of the chord $PQ$.

   [1 mark]

   • A$(11,\,7)$
   • B$(5.5,\,3.5)$
   • C$(0.5,\,3.5)$
   • D$(3.5,\,5.5)$

   (b)

   The line $CM$ passes through the centre $C$ and the midpoint $M$ of $PQ$. Find its gradient.

   [1 mark]

   • A$7$
   • B$-\frac17$
   • C$-7$
   • D$\frac17$

   (c)

   Find the exact distance from $C$ to the chord $PQ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The circle $S$ has equation $(x-1)^2+(y+2)^2=25$. The point $T=(4,2)$ lies on $S$.

   (a)

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

   [3 marks]

   (b)

   The tangent to $S$ at $T$ meets the $x$-axis at $A$ and the $y$-axis at $B$. Find the area of triangle $OAB$, where $O$ is the origin.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The circle $C$ has centre $(4,3)$ and equation $x^2+y^2-8x-6y+9=0$. The line $l$ has equation $y=x-3$ and meets $C$ at the points $P$ and $Q$.

   (a)

   (i) Show that the $x$-coordinates of $P$ and $Q$ satisfy $x^2-10x+18=0$.
   (ii) Use a circle property to find the coordinates of the midpoint $M$ of $PQ$ without solving this equation.

   [6 marks]

   (b)

   Given that the midpoint of $PQ$ is $M=(5,2)$, find (i) the exact length of $PQ$, (ii) the area of triangle $CPQ$.

   [6 marks]

   Total for question 4: 12 marks

End of questions