Pure C: Coordinate Geometry & Curves

Topic overview

Parametric equations, polar coordinates, area and arc length

Key concepts

Verify STEP 2/3 separately. Complex numbers and 2×2 matrices may belong to STEP 2; De Moivre, general complex roots, planes and vector products, etc. belong to STEP 3.

Key points

◆Follow the prerequisite knowledge requirements given by the examination body; new concepts may only be used when the question provides sufficient guidance.
◆The examples on this site are training with a limited scope and do not represent full syllabus coverage or official difficulty.

Worked examples

Try each example yourself before checking the explanation.

1Written examples within the scope

The curve $C$ is defined parametrically by $x = t - \sin t$ , $y = 1 - \cos t$ for $t \in [0,\, 2\pi]$ .

Find $\dfrac{dy}{dx}$ in terms of $t$ . At what values of $t$ is the tangent horizontal?

Show that the length of arc $C$ is $8$ .

Find the area enclosed between $C$ and the $x$ -axis.

By substituting $t=2u$ , or otherwise, show that $\displaystyle\int_0^{2\pi}\sin^2\!\left(\frac t2\right)dt=\pi$ . Hence show that $\displaystyle\int_0^{2\pi}y\,dt=2\pi$ .

Ready to practise this topic?

Practise the available questions for this topic. Difficulty labels are platform references.

Practise Pure C: Coordinate Geometry & Curves →