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Pure · Coordinate geometry

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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.

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

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 tt for as long as you can.

Meeting the axes. A point is on the xx-axis when y=0y = 0, and on the yy-axis when x=0x = 0. So set the relevant coordinate's formula to zero and solve for tt:
  • xx-axis: solve g(t)=0g(t) = 0 (i.e. y=0y = 0), then substitute each root into x=f(t)x = f(t).
  • yy-axis: solve f(t)=0f(t) = 0 (i.e. x=0x = 0), then substitute each root into y=g(t)y = g(t).
For x=t+3, y=t2−4x = t + 3,\ y = t^2 - 4: the curve meets the xx-axis where t2−4=0t^2 - 4 = 0, i.e. t=±2t = \pm 2, giving (5,0)(5, 0) and (1,0)(1, 0); it meets the yy-axis where t+3=0t + 3 = 0, i.e. t=−3t = -3, giving (0,5)(0, 5).

Meeting a line y=mx+cy = mx + c. Substitute the parametric xx and yy straight into the line and solve the resulting equation in tt. Each solution is one crossing point. For x=t2, y=2tx = t^2,\ y = 2t meeting y=x−3y = x - 3: 2t=t2−3 ⇒ t2−2t−3=0 ⇒ (t−3)(t+1)=0,2t = t^2 - 3 \ \Rightarrow\ t^2 - 2t - 3 = 0 \ \Rightarrow\ (t-3)(t+1) = 0, so t=3t = 3 or t=−1t = -1, giving the points (9,6)(9, 6) and (1,−2)(1, -2).

A repeated root means a tangent. If the equation in tt is a quadratic with a single (repeated) root, the line touches the curve at exactly one point — it is a tangent. Seeing (t−1)2=0(t - 1)^2 = 0 rather than two distinct roots is the signal.

Why stay in tt? You could convert to Cartesian first and solve in xx, but that often loses or doubles solutions when the relationship is many-to-one (think of a curve that passes through a point twice for two different tt). Solving for the parameter keeps every crossing, each labelled by its own tt.