Equation of a Circle (Centre (h,k))

Statement

The equation of a circle with centre at \((h,k)\) and radius \(r\) is:

\[ (x-h)^2 + (y-k)^2 = r^2 \]

This represents all points \((x,y)\) that are exactly distance \(r\) from the centre \((h,k)\).

Why it’s true

• The distance between two points \((x,y)\) and \((h,k)\) is given by the distance formula: \(\sqrt{(x-h)^2+(y-k)^2}\).
• On the circle, this distance is always equal to the radius \(r\).
• So, \(\sqrt{(x-h)^2+(y-k)^2}=r\). Squaring gives \((x-h)^2+(y-k)^2=r^2\).

Recipe (how to use it)

1. Identify the centre \((h,k)\) and radius \(r\) of the circle.
2. Substitute into \((x-h)^2+(y-k)^2=r^2\).
3. To check if a point lies on the circle, substitute its coordinates and see if the equation holds.

Spotting it

This form is used whenever a circle is not centred at the origin. The terms \((x-h)\) and \((y-k)\) show the horizontal and vertical shift of the circle.

Common pairings

• Distance formula (basis of derivation).
• Equation of a circle at the origin: \(x^2+y^2=r^2\).
• Circle geometry problems involving translations.

Mini examples

1. Equation: \((x-2)^2+(y-3)^2=25\). Centre: (2,3), Radius: 5.
2. Equation: \((x+1)^2+(y-4)^2=9\). Centre: (-1,4), Radius: 3.
3. Equation: \((x-0)^2+(y+2)^2=16\). Centre: (0,-2), Radius: 4.

Pitfalls

• Forgetting the minus signs: if the equation is \((x-h)^2+(y-k)^2=r^2\), then the centre is \((h,k)\), not \((-h,-k)\).
• Mixing up radius \(r\) with \(r^2\).

Exam strategy

• Always expand carefully if you need the circle in standard form.
• Check the sign: if it’s \((x+3)^2\), the centre x-coordinate is \(-3\).
• Use substitution to test whether a point lies inside, outside, or on the circle.

Summary

The equation \((x-h)^2+(y-k)^2=r^2\) describes a circle with centre \((h,k)\) and radius \(r\). It generalises the origin-centred case and is central to coordinate geometry.