Equation of a Circle from a Diameter

Two diameter endpoints hand you everything: the centre is their midpoint, and the radius is half the distance between them. Feed both into the standard form and the equation follows. The one thing to watch is halving the diameter to get the radius.

The points $A(1, 2)$ and $B(7, 10)$ are the ends of a diameter of a circle. Find the equation of the circle. [5]

The working

Step 1, centre = midpoint of $AB$: $\left(\frac{1 + 7}{2}, \ \frac{2 + 10}{2}\right) = (4, 6) \quad \text{(M1, A1)}$

Step 2, radius = half the length of $AB$. First the length: $AB = \sqrt{(7 - 1)^2 + (10 - 2)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \quad \text{(M1)}$

So the radius is $r = \frac{1}{2}(10) = 5$.

Step 3, substitute into $(x - a)^2 + (y - b)^2 = r^2$ with centre $(4, 6)$ and $r = 5$: $(x - 4)^2 + (y - 6)^2 = 25 \quad \text{(A1)}$

A useful check: the radius also equals the distance from the centre $(4,6)$ to $A(1,2)$: $\sqrt{3^2 + 4^2} = 5$ ✓.

Where the marks are won and lost

• The radius is half of $AB$. Using $r = 10$ (the diameter) gives $r^2 = 100$ and a wrong circle, this is the single most common error.
• The right-hand side is $r^2 = 25$, not $r = 5$.
• Centre signs go straight in as $(4, 6)$, giving brackets $(x - 4)$ and $(y - 6)$.

Common mistakes

• Writing $r^2 = 100$ by using the full diameter as the radius.
• Midpoint slips (adding but forgetting to divide by $2$).
• Putting $r = 5$ on the right instead of $r^2 = 25$.

Full method: Equation of a Circle notes. See also Gradient, Midpoint & Length. Topic home: Circle Geometry pillar.