The equation of a circle is a fundamental concept in coordinate geometry, essential for solving various geometrical problems. We will explore the standard and general forms of a circle's equation, focusing on finding the centre and radius and converting between these forms.

**Standard Form of a Circle's Equation**

**Equation**: $(x-a)^2 + (y-b)^2 = r^2$**Centre**: $(a, b)$**Radius**: $r$

In the standard form, $a$ and $b$ represent the coordinates of the centre of the circle, and $r$ is the radius.

**General Form of a Circle's Equation**

**Equation**: $x^2 + y^2 + 2gx + 2fy + c = 0$**Centre**:$\left(-g, -f\right)$**Radius**: $\sqrt{g^2 + f^2 - c}$

**Key Concepts**

**Completing the Square**: A method to convert the general form of a circle's equation to the standard form.**Tangents and Radius**: Tangents to a circle are always perpendicular to the radius at the point of contact.**Right-Angled Triangle in Circle**: If a right-angled triangle is inscribed in a circle, its hypotenuse is the diameter of the circle.

**Example 1**

Given the equation of a circle $x^2 + y^2 + 6x - 8y + 9 = 0$, find the centre and radius.

**Solution**:

1. Convert to standard form:

$x^2 + 6x + y^2 - 8y = -9$

$(x + 3)^2 - 9 + (y - 4)^2 - 16 = -9$

$(x + 3)^2 + (y - 4)^2 = 16$

2. Centre: $(-3, 4)$

3. Radius: $\sqrt{16} = 4$

**Example 2**

Find the centre and radius of the circle given by the equation $x^2 + y^2 - 12x + 4y + 36 = 0$.

**Solution**:

1. Convert to standard form:

$x^2 - 12x + y^2 + 4y = -36$

$(x - 6)^2 - 36 + (y + 2)^2 - 4 = -36$

$(x - 6)^2 + (y + 2)^2 = 4$

2. Centre: $(6,-2)$

3. Radius: $\sqrt{4} = 2$

**Example 3**

**Task**: A circle has the equation $x^2 + y^2 - 4x + 6y - 12 = 0$. Determine its centre and radius.

**Solution**:

1. Rearrange to standard form:

$x^2 - 4x + y^2 + 6y = 12$

$(x - 2)^2 - 4 + (y + 3)^2 - 9 = 12$

$(x - 2)^2 + (y + 3)^2 = 25$

2. Centre: $(2,-3)$

3. Radius: $\sqrt{25} = 5$

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.