About this chapter
In Pure 4, coordinate geometry means parametric equations: x and y each given in terms of a third variable t. The central skill is eliminating t to get a Cartesian equation, and the October 2022 examiners described students who only partly rearranged and ended with y in terms of both x and t.
This chapter covers finding points from t, eliminating t by substitution, using Pythagorean, double-angle and addition identities, exponentials and logs, domain and range, sketching curves that cross themselves, where a curve meets a line, and a running track modelled parametrically.
The ten questions
- Q1 (5 marks): show every point lies on a straight line, then decide whether the curve meets x + y = 13.
- Q2 (5 marks): eliminate t using tan and sec² to get a(x + b)² + c, then the domain and range.
- Q3 (5 marks): x = 2 sin t, y = sin 2t gives 4y² = x²(4 − x²), and where it meets the x-axis.
- Q4 (5 marks): expand a cos(t + π/6) and show every point satisfies x² + xy + y² = k.
- Q5 (5 marks): x = ln(t + 1) gives a Cartesian equation in powers of e, its x-axis crossings and range.
- Q6 (11 marks): a curve that crosses itself, its Cartesian equation, and where lines through the origin meet it.
- Q7 (9 marks): part of a circle: centre and radius, a sketch, and when y = x + k meets it twice.
- Q8 (10 marks): a runner on an elliptical track: its equation, the distance from the centre, and the times at a given distance.
- Q9 (10 marks): x = tan t, y = cos 2t as a rational function, then a proof that y never reaches −1.
- Q10 (11 marks): an astroid: a Cartesian equation, x² + y² in terms of sin 2t, and its nearest and furthest points from O.
Key skills tested
Points from t. Substitute t to find a point. For a condition such as y = 0, solve for t first, then find x.
Eliminating t. Make t the subject of one equation and substitute into the other. For example, x(t+1) = 2t gives t = x/(2 − x). The final equation has no t in it.
Pythagorean identities. sin²t + cos²t = 1 and 1 + tan²t = sec²t. x = a + r cos t, y = b + r sin t is a circle.
Double angles and addition. sin 2t = 2 sin t cos t and cos 2t = 1 − 2 sin²t. Expand cos(t + π/6) first, then eliminate t.
Exponentials and logs. x = ln(t + 1) means t = e to the power x, minus 1. Write the answer in the form asked for.
Domain and range. The domain comes from the values of x(t) and the range from y(t). Check turning points inside the interval, not only the ends.
Curve sketching. Use key values of t. Two values of t giving the same point means the curve crosses itself.
Meeting a line. Substitute x(t) and y(t) into the line to get an equation in t, and reject values outside the interval.
Modelling. Say what t, x and y mean, and give answers in context with units and a limitation.
Worked example
Question 1 from this chapter. C has x = 2t/(t+1) and y = (t − 3)/(t+1), for t > −1. (a) Show that every point on C lies on a straight line, and find it. (b) Determine whether C meets x + y = 13.
(a) From x(t+1) = 2t, t(x − 2) = −x, so t = x/(2 − x). Substituting, t − 3 = (4x − 6)/(2 − x) and t + 1 = 2/(2 − x), so y = (4x − 6)/2 = 2x − 3. Every point lies on y = 2x − 3.
(b) On the line, x + y = 13 gives 3x − 3 = 13, so x = 16/3. But x = 2 − 2/(t+1), which is less than 2 whenever t > −1, so x can never be 16/3. C does not meet the line.
The reason in (b) comes from the restriction on t, not from the line alone.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- October 2022, Q1: eliminating t from a rational parametric equation, as in Q1.
- January 2024, Q9(c): a Cartesian equation from parametric equations with trig terms, using compound-angle and Pythagorean identities, as in Q2 and Q4.
Where marks are lost
- t still in the answer. In October 2022 a disappointing number rearranged only part of the way and ended with y in terms of both x and t.
- Stopping at the constants. In January 2024 some found the values of A and B from points on the curve without showing the equation had that form, and gained nothing.
- The algebra at the end. Rearranging into the exact form asked for, including rationalising, was where only the most able succeeded in January 2024.
- Ignoring the interval. A value of t outside the given interval does not give a point on the curve.
Common questions
What makes an equation Cartesian?
It contains only x and y, with the parameter t completely eliminated.
How do I choose which equation to rearrange?
Pick the simpler one to make t the subject, or use an identity when both involve trig functions of t.
How do I find the range of a parametric curve?
Find the values y takes as t runs over its interval, checking any turning points as well as the end values.
Where this chapter leads
- Pure 4 Chapter 5: Differentiation: gradients, tangents and normals of parametric curves.
- Pure 4 Chapter 6: Integration: areas and volumes under parametric curves.