Parametric equationsEdexcel A-Level Maths: Revision notes

In parametric equations the coordinates $x$ and $y$ are each given in terms of a third variable, the parameter, usually $t$ or $\theta$. For each value of the parameter you get one point $(x,y)$, and as the parameter changes the point traces out a curve. Example: $x=2t+1$, $y=4t^2-3$. When $t=2$, $(x,y)=(5,13)$. To find where the curve meets an axis, set $x=0$ (the $y$-axis) or $y=0$ (the $x$-axis), solve for $t$, then substitute back to find the other coordinate. To find where it meets a line, substitute both parametric equations into the line's equation and solve for $t$.

To get a Cartesian equation (one linking $x$ and $y$ only), eliminate the parameter.

• Substitution: rearrange one equation for $t$ and substitute into the other. From $x=2t+1$, $t=\frac{x-1}{2}$, so $y=4\left(\frac{x-1}{2}\right)^2-3=x^2-2x-2$.
• Multiplying or dividing: for $x=5t$, $y=\frac5t$, the product $xy=25$ removes $t$ directly.
• Trigonometric identity: if $x$ and $y$ involve $\cos t$ and $\sin t$, rearrange to get $\cos t$ and $\sin t$ alone and use $\cos^2t+\sin^2t=1$. Useful forms: $x=5t$, $y=3t^2$ gives $25y=3x^2$ (a parabola); $x=5t$, $y=\frac5t$ gives $xy=25$ (a hyperbola).

The equations $x=r\cos t$, $y=r\sin t$ give the circle $x^2+y^2=r^2$, since $r^2\cos^2t+r^2\sin^2t=r^2$. For example $x=3\cos t$, $y=3\sin t$ is a circle of radius $3$ about the origin. A shifted circle is $x=a+r\cos t$, $y=b+r\sin t$, with centre $(a,b)$ and radius $r$: $\cos t=\frac{x-a}{r},\ \sin t=\frac{y-b}{r}\ \Rightarrow\ (x-a)^2+(y-b)^2=r^2.$ The examples $x=2+5\cos t$, $y=-4+5\sin t$ give centre $(2,-4)$ and radius $5$. The point starts at the right-hand end of the horizontal diameter when $t=0$ and moves anticlockwise as $t$ increases.

The values allowed for $t$ decide which part of the curve is drawn, so always read the domain.

• $0\le t<2\pi$ gives a full circle, once round.
• $0\le t\le\pi$ for $x=2+4\cos t$, $y=-1+4\sin t$ gives only $y\ge-1$, the upper semicircle.
• For $x=\frac{12}{t}$-type curves, $t\ne0$ means the Cartesian form has the restriction $x\ne0$.
• A restricted domain such as $-2\le t\le3$ on $x=4t$, $y=3t^2$ gives only part of the parabola $16y=3x^2$, with $-8\le x\le12$. When you solve for $t$, reject any value outside the allowed range. Trigonometric equations usually give two values in $[0,2\pi)$: for $\cos t=\frac12$, $t=\frac{\pi}{3}$ and $\frac{5\pi}{3}$.