The Equation of a Circle
Master the equation of a circle from first principles — derivation from the distance formula, completing the square, determining point positions, and finding equations with fully worked examples.
Analytical Geometry
Table of Contents
Analytical Geometry: The Circle (~40 marks, Paper 2)#
Analytical Geometry is worth ~40 marks in Paper 2. In Grade 12, the focus shifts from straight lines to the Equation of a Circle.
The Big Idea: Pythagoras in a Loop#
The equation of a circle is just the distance formula (Pythagoras) in disguise. If every point $(x; y)$ on the edge of the circle is exactly $r$ (the radius) distance away from the centre $(a; b)$, then:
$$ (x - a)^2 + (y - b)^2 = r^2 $$Special case: Centre at the origin → $x^2 + y^2 = r^2$.
The Two Forms#
| Form | Equation | What you can read |
|---|---|---|
| Standard | $(x - a)^2 + (y - b)^2 = r^2$ | Centre = $(a; b)$, radius = $r$ |
| General | $x^2 + y^2 + Dx + Ey + F = 0$ | Must complete the square to find centre and radius |
Converting General → Standard#
- Group $x$-terms and $y$-terms: $(x^2 + Dx) + (y^2 + Ey) = -F$
- Complete the square for each: add $(\frac{D}{2})^2$ and $(\frac{E}{2})^2$ to both sides
- Read off centre and radius
The Tangent Strategy#
A tangent touches the circle at exactly one point. The key relationship:
$$\text{radius} \perp \text{tangent at the point of contact}$$To find the tangent equation:
- Find the gradient of the radius: $m_r = \frac{y_1 - b}{x_1 - a}$
- Tangent gradient = negative reciprocal: $m_t = -\frac{1}{m_r}$
- Use point-gradient form: $y - y_1 = m_t(x - x_1)$
The Toolkit You Need (Grade 10–11 Revision)#
| Tool | Formula | Use |
|---|---|---|
| Distance | $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ | Finding radius, proving equal lengths |
| Midpoint | $M = \left(\frac{x_1 + x_2}{2};\, \frac{y_1 + y_2}{2}\right)$ | Finding centre from diameter endpoints |
| Gradient | $m = \frac{y_2 - y_1}{x_2 - x_1}$ | Radius gradient for tangent problems |
| Parallel | $m_1 = m_2$ | Parallel tangent problems |
| Perpendicular | $m_1 \times m_2 = -1$ | Radius ⊥ tangent |
| Line equation | $y - y_1 = m(x - x_1)$ | Writing tangent equations |
Determining Position Relative to a Circle#
Substitute the point $(x; y)$ into the LHS of $(x - a)^2 + (y - b)^2$:
| Result | Position |
|---|---|
| $< r^2$ | Inside the circle |
| $= r^2$ | On the circle |
| $> r^2$ | Outside the circle |
Deep Dives (click into each)#
- The Equation of a Circle — standard form, shifted centres, completing the square, and inside/on/outside tests
- Tangents to Circles — the radius ⊥ tangent strategy, finding tangent equations, and worked exam-style problems
🚨 Common Mistakes#
- Signs in the equation: $(x - 3)^2 + (y + 2)^2 = 25$ has centre $(3;\, -2)$, NOT $(3;\, 2)$. The signs flip!
- Completing the square errors: Add the completing values to BOTH sides of the equation.
- Tangent gradient: It’s the negative reciprocal of the radius gradient, not just the reciprocal.
- Not checking the point is on the circle: Before finding a tangent at a point, verify the point satisfies the circle equation.
- $r^2$ vs $r$: The equation gives $r^2$. The radius is $r = \sqrt{r^2}$. Don’t forget the square root.
- Forgetting to state the radius is > 0: After completing the square, if $r^2 \leq 0$, the equation does NOT represent a circle.
🔗 Related topics:
- Euclidean Geometry — geometric circle properties complement the algebraic approach
- Trigonometry — angle of inclination connects gradients to trig
📌 Grade 10/11 foundations:
- Core Formulas — distance, midpoint, gradient
- Inclination, Circles & Tangents — Grade 11 circle equation introduction
⏮️ Trigonometry | 🏠 Back to Grade 12 | ⏭️ Euclidean Geometry
Tangents to Circles
Master finding tangent equations to circles — the perpendicular radius principle, multiple exam scenarios, tangents from external points, and the length of a tangent with fully worked examples.