Equation of a Circle Touching the x-axis

A circle touches a line when that line is a tangent, so the distance from the centre to the line equals the radius. For the $x$-axis, that distance is simply the size of the centre’s $y$-coordinate.

A circle has centre $(3, 4)$ and touches the $x$-axis. Find the equation of the circle. [3]

The working

Step 1, find the radius. The distance from the centre $(3, 4)$ to the $x$-axis is the $y$-coordinate: $r = 4 \quad \text{(M1)}$

(The circle just reaches down to touch $y = 0$ at the point $(3, 0)$.)

Step 2, write the equation in the form $(x - a)^2 + (y - b)^2 = r^2$ with centre $(3, 4)$: $(x - 3)^2 + (y - 4)^2 = 4^2 \quad \text{(M1)}$

Step 3, simplify $r^2$: $(x - 3)^2 + (y - 4)^2 = 16 \quad \text{(A1)}$

Where the marks are won and lost

• “Touches the $x$-axis” $\Rightarrow$ radius $=$ distance to the $x$-axis $=$ the $y$-coordinate of the centre, here $4$.
• Square the radius in the equation: the right side is $r^2 = 16$, not $4$.
• Keep the centre’s signs correct: centre $(3,4)$ gives $(x-3)^2 + (y-4)^2$.

Common mistakes

• Using $r = 3$ (the $x$-coordinate) instead of $r = 4$.
• Writing $= 4$ instead of $= 16$ on the right.
• Confusing “touches the $x$-axis” (radius $=$ $y$-coordinate) with “touches the $y$-axis” (radius $=$ $x$-coordinate).

Topic home: Coordinate Geometry of the Circle pillar. More: Worked examples.