Polar coordinates and polar curvesEdexcel A-Level Further Maths: Subtopic test

1. 1
   The point $P$ has polar coordinates $\left(4,\frac{2\pi}{3}\right)$ and the point $Q$ has Cartesian coordinates $(-1,-1)$.

   (a)

   Find the $x$-coordinate of $P$.

   [1 mark]

   • A$2$
   • B$2\sqrt3$
   • C$-2\sqrt3$
   • D$-2$

   (b)

   Find the $y$-coordinate of $P$.

   [1 mark]

   • A$2\sqrt3$
   • B$-2$
   • C$-2\sqrt3$
   • D$2$

   (c)

   Find the polar coordinates $(r,\theta)$ of $Q$, with $r>0$ and $-\pi<\theta\le\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has polar equation $r=4\cos\theta$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

   (a)

   Which is the Cartesian equation of $C$?

   [1 mark]

   • A$x^2+y^2=16$
   • B$(x-4)^2+y^2=16$
   • C$(x-2)^2+y^2=4$
   • D$x^2+(y-2)^2=4$

   (b)

   Find the Cartesian coordinates of the point of $C$ where $\theta=\frac{\pi}{3}$.

   [1 mark]

   • A$(2,\sqrt3)$
   • B$(1,\sqrt3)$
   • C$(\sqrt3,1)$
   • D$(1,2)$

   (c)

   The circle with polar equation $r=2$ meets $C$ at two points. Find their polar coordinates, with $r>0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has polar equation $r=2(1+\cos\theta)$ for $-\pi<\theta\le\pi$.

   (a)

   (i) Show that $C$ is symmetrical about the initial line.
   (ii) State the greatest value of $r$ and the value of $\theta$ at which it occurs.

   [3 marks]

   (b)

   The points $A$ and $B$ on $C$ have $\theta=\frac{\pi}{3}$ and $\theta=\frac{2\pi}{3}$ respectively. Find the exact distance $AB$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has polar equation $r^2=9\cos2\theta$.

   (a)

   (i) Show that the Cartesian equation of $C$ is $(x^2+y^2)^2=9(x^2-y^2)$.
   (ii) Hence explain why every point of $C$ satisfies $|y|\le|x|$.

   [6 marks]

   (b)

   The line $L$ has polar equation $r=3\sec\theta$. Show that $L$ meets $C$ at exactly one point, and find the polar coordinates of that point.

   [6 marks]

   Total for question 4: 12 marks

End of questions