Circles, loci and regions: practice problems

Equations of circles

To read off the centre and radius from the general form, complete the square in $x$ and in $y$. For a circle through three points, substitute the points into the general form and solve for $l$, $m$, $n$.

Circles and lines; tangents

Let $d$ be the distance from the centre to a line and $r$ the radius: the line cuts the circle twice if $d < r$, touches it if $d = r$, and misses it if $d > r$.

A chord at distance $d$ from the centre has length $2\sqrt{r^2 - d^2}$. Subtracting the equations of two intersecting circles gives the line through their intersection points.

Loci

To find a locus, let P $= (x,\ y)$, write the condition as an equation in $x$ and $y$, and identify the curve.

Example: points P with $\mathrm{PA} : \mathrm{PB} = 2 : 1$ for A$(0,\ 0)$, B$(3,\ 0)$ satisfy $\mathrm{PA}^2 = 4\mathrm{PB}^2$, i.e. $(x - 4)^2 + y^2 = 4$: a circle with centre $(4,\ 0)$ and radius $2$ (a circle of Apollonius).

If P depends on a moving point Q $= (s,\ t)$, express $s$ and $t$ in terms of $x$ and $y$ and substitute into the condition on Q.

Regions and maximum or minimum values

$y > mx + n$ is the region above a line, and $(x - a)^2 + (y - b)^2 < r^2$ is the inside of a circle.

To maximise $ax + by$ over a region, set $ax + by = k$: this is a line, and you want the largest $k$ for which it still meets the region. For a polygonal region the extreme values occur at vertices.