Coordinate geometry: lines and circlesAQA A-Level Maths: Topic test

1. 1
   The line $l$ passes through the points $P(-2,7)$ and $Q(4,-5)$.

   (a)

   What is the gradient of $l$?

   [1 mark]

   • A$-2$
   • B$2$
   • C$-\dfrac{1}{2}$
   • D$\dfrac{1}{2}$

   (b)

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

   [1 mark]

   • A$-2$
   • B$-\dfrac{1}{2}$
   • C$\dfrac{1}{2}$
   • D$2$

   (c)

   Find an equation of $l$ 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 centre $(-3,4)$ and radius $\sqrt{18}$.

   (a)

   Which is an equation of $C$?

   [1 mark]

   • A$(x-3)^2+(y+4)^2=18$
   • B$(x+3)^2+(y-4)^2=18$
   • C$(x+3)^2+(y-4)^2=\sqrt{18}$
   • D$(x+3)^2+(y-4)^2=36$

   (b)

   Where does the point $(0,8)$ lie in relation to $C$?

   [1 mark]

   • Ainside $C$
   • Bon $C$
   • Cat the centre of $C$
   • Doutside $C$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The circle $C$ has equation $x^2+y^2+4x-6y-12=0$ and the line $l$ has equation $y=x$.

   (a)

   Find the coordinates of the centre of $C$ and the radius of $C$.

   [3 marks]

   (b)

   The line $l$ meets $C$ at two points. Find their coordinates and the length of the chord between them, giving your answer in surd form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The circle $C$ has centre $M(3,-1)$ and passes through the point $T(7,2)$. The line $t$ is the tangent to $C$ at $T$.

   (a)

   Find the equation of $C$ and show that the equation of $t$ is $4x+3y=34$.

   [6 marks]

   (b)

   The point $S$ is at the opposite end of the diameter through $T$. The tangent to $C$ at $S$ is $t'$. Find the coordinates of $S$ and the equation of $t'$ in the form $4x+3y=k$. Explain why the distance between $t$ and $t'$ is 10.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The circle $C$ has equation $x^2+y^2-10x+4y+13=0$.

   (a)

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

   [1 mark]

   • A$(-5,2)$
   • B$(5,2)$
   • C$(10,-4)$
   • D$(5,-2)$

   (b)

   What is the radius of $C$?

   [1 mark]

   • A$16$
   • B$4$
   • C$\sqrt{42}$
   • D$\sqrt{13}$

   (c)

   The point $(5,2)$ lies on $C$. Find the equation of the tangent to $C$ at this point.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The straight line $l_1$ has equation $2y=6x-4$ and the straight line $l_2$ has equation $3y+x=9$.

   (a)

   What is the gradient of $l_1$?

   [1 mark]

   • A$6$
   • B$-3$
   • C$3$
   • D$-2$

   (b)

   Which statement about $l_1$ and $l_2$ is correct?

   [1 mark]

   • AThey are perpendicular.
   • BThey are parallel.
   • CThey are neither parallel nor perpendicular.
   • DThey are the same line.

   (c)

   Find the coordinates of the point where $l_1$ and $l_2$ intersect.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A water tank is filled at a constant rate. The depth of water is 12 cm after 3 minutes and 27 cm after 8 minutes. The tank is 60 cm deep. The depth $d$ cm is modelled as a linear function of the time $t$ minutes after filling began.

   (a)

   Find an equation for $d$ in terms of $t$.

   [3 marks]

   (b)

   Use the model to find when the tank is full. Interpret the gradient and the constant term in the context of the question, and state one limitation of the model.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The triangle $PQR$ has vertices $P(1,3)$, $Q(9,7)$ and $R(3,-1)$.

   (a)

   Show that angle $QPR$ is a right angle. Hence find the equation of the circle that passes through $P$, $Q$ and $R$.

   [6 marks]

   (b)

   The line through $R$ parallel to $PQ$ meets the circle through $P$, $Q$ and $R$ again at the point $S$. Find the coordinates of $S$.

   [6 marks]

   Total for question 8: 12 marks

End of questions