8.4PureStretch

Points of Intersection

To find where a parametric curve meets an axis or another curve, substitute the condition into the parametric equations, solve for the parameter, then evaluate the coordinates.

26 min Parametric Equations

Edexcel A level Maths: 8.4 Points of IntersectionCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Find axis crossings of a parametric curve
Find intersections with a given line
Solve for the parameter value(s)
Recover the coordinates

Crossing the axes

The curve crosses the $x$ -axis where $y = 0$ , and the $y$ -axis where $x = 0$ . Solve the relevant equation for $t$ , then substitute back to get the point.

Example 1Where does

x = t^2 - 4

y = 2t

cross the

x

-axis?

x

-axis:

y = 0 \Rightarrow 2t = 0 \Rightarrow t = 0

x = 0^2 - 4 = -4

Answer

(-4, 0)

Tip — x-axis means y = 0; y-axis means x = 0 — solve for t first.

Intersection with a line

Substitute the parametric $x$ and $y$ into the line’s equation to get an equation in $t$ . Solve for $t$ , then find each intersection point.

Formula recap

y = 0 \text{ for } x\text{-axis}, \ x = 0 \text{ for } y\text{-axis}

Axis crossings.

g(t) = \text{line in } f(t)

Substitute to find t.

Common mistakes to avoid

Reading the coordinate value as the parameter.

Solve for t, then substitute back to get the actual point.

Missing a second solution for t.

A quadratic in t can give two intersection points.

Key takeaways

x-axis: set y = 0; y-axis: set x = 0.
Solve for t, then substitute to find the point.
For a line, substitute parametric x, y into its equation.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A parametric curve crosses the x-axis where:
- $y = 0$
- $x = 0$
- $t = 0$
- $y = x$
Show answer and explanation
Answer: $y = 0$
On the x-axis y is zero.
For $x = t^2 - 4$ , $y = 2t$ , the x-axis crossing is at t =
- $0$
- $2$
- $-4$
- $4$
Show answer and explanation
Answer: $0$
2t = 0.
That crossing point is:
- $(-4, 0)$
- $(0, 0)$
- $(4, 0)$
- $(0, -4)$
Show answer and explanation
Answer: $(-4, 0)$
x = 0² − 4.
To intersect a curve with a line you:
- substitute parametric x, y into the line
- differentiate
- add equations
- plot only
Show answer and explanation
Answer: substitute parametric x, y into the line
Gives an equation in t.
A quadratic in t for intersections may give:
- two points
- no t
- a gradient
- an asymptote
Show answer and explanation
Answer: two points
Two roots.

Next step

Check your understanding

Work through questions on Points of Intersection. Use the feedback to decide what needs another look.

5 past-paper questions · 15 marks

Practise this topic

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