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 Video by Zeeshan Zamurred Parametric Equations

Edexcel A level Maths: 8.4 Points of IntersectionWatch the full walkthrough 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.

x

-axis:

y = 0 \Rightarrow 2 t = 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 for x -axis, x = 0 for y -axis

Axis crossings.

g (t) = 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

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