The idea

Intersections of parametric curves

Finding where a parametric curve meets the coordinate axes, a given line, or another curve — by solving for the parameter — and using a repeated root to detect a tangent.

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Pure · Coordinate geometry

Intersections of parametric curves

Finding where a parametric curve meets the coordinate axes, a given line, or another curve — by solving for the parameter — and using a repeated root to detect a tangent.

Why it works

Solve for t, then read off the point

Every intersection question reduces to the same idea: find the value(s) of the parameter at the crossing point, then read off the coordinates. Work in

t

for as long as you can.

Meeting the axes

A point is on the

x

-axis when

y = 0

, and on the

y

-axis when

x = 0

. So set the relevant coordinate's formula to zero and solve for

t

$x$ -axis: solve $g(t) = 0$ (i.e. $y = 0$ ), then substitute each root into $x = f(t)$ .
$y$ -axis: solve $f(t) = 0$ (i.e. $x = 0$ ), then substitute each root into $y = g(t)$ .

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The rest of the explanation, plus 3 worked examples you step through move by move.

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