Circle properties and tangentsAQA A-Level Maths: Subtopic test

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

   (a)

   Find the gradient of the radius $CA$.

   [1 mark]

   • A$\frac{4}{3}$
   • B$-\frac{3}{4}$
   • C$\frac{3}{5}$
   • D$\frac{3}{4}$

   (b)

   Find the gradient of the tangent to the circle at $A$.

   [1 mark]

   • A$-\frac{4}{3}$
   • B$\frac{3}{4}$
   • C$\frac{4}{3}$
   • D$-\frac{3}{4}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $AB$ is a diameter of a circle, where $A(-1,1)$ and $B(7,5)$.

   (a)

   Find the centre of the circle.

   [1 mark]

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

   (b)

   Which of these points $P$ gives $A\hat{P}B=90^\circ$?

   [1 mark]

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

   (c)

   Find the equation of the circle with diameter $AB$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A circle has centre $O(2,3)$ and radius $5$. The points $P(-1,7)$ and $Q(6,0)$ lie on the circle, and $M$ is the midpoint of the chord $PQ$.

   (a)

   Find the coordinates of $M$ and show that $OM$ is perpendicular to $PQ$.

   [3 marks]

   (b)

   Use the fact that $OM$ is perpendicular to the chord $PQ$, together with Pythagoras in triangle $OMP$, to confirm that the radius of the circle is $5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A circle $C$ has equation $(x-1)^2+(y+2)^2=40$. The point $T(7,-4)$ lies on $C$, and $K$ is the centre of $C$.

   (a)

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

   [6 marks]

   (b)

   The tangent at $T$ meets the $x$-axis at $P$. Find the coordinates of $P$ and the exact length of $PT$.

   [6 marks]

   Total for question 4: 12 marks

End of questions