Equation of a Circle

A circle is the locus of a moving point in a plane that always remains at a constant distance from a fixed point in the same plane.

The fixed point is called the centre of the circle, and the constant distance is called its radius.

Suppose the centre of the circle is O and its radius is r. For any point P on the circle,

OP = r

When this geometric condition is expressed using coordinates, we obtain the equation of the circle.

👉 Equation of a Circle Centred at the Origin

Suppose the centre of a circle in the Cartesian plane is the origin O(0, 0), and P(x, y) is any point on the circle. Let the radius of the circle be a.

By the Distance Formula,

But OP = a.

Therefore,

Squaring both sides,

x² + y² = a²

This is the standard equation of a circle with centre at the origin and radius a.

Example

If a circle has centre (0, 0) and radius 5 units, its equation is

Therefore,

x² + y² = 25

👉 Equation of a Circle with Centre (α, β)

Suppose the centre of a circle is Q(α, β), P(x, y) is any point on the circle, and the radius is a.

Then,

Using the Distance Formula,

Squaring both sides,

(x - α)² + (y - β)² = a²

This is the standard form of a circle with centre (α, β) and radius a.

Remember

From the equation (x - α)² + (y - β)² = a², we can directly identify:

• Centre = (α, β)
• Radius = a

Example

Here,

• Centre = (3, -2)
• Radius = 4

because (y + 2) = (y - (-2)).

👉 General Equation of a Circle

The standard equation of a circle is

Expanding,

This is commonly written as

x² + y² + 2gx + 2fy + c = 0

Comparing the two forms,

g = -α and f = -β

Therefore, the centre of the circle is

(-g, -f)

and its radius is

r = √(g² + f² - c)

For the equation to represent a real circle with positive radius,

must hold.

Example

Consider the circle

Here,

and c = -3.

Therefore, the centre is

and the radius is

Thus, the circle has centre (3, -2) and radius 4 units.

👉 Finding the Centre and Radius by Completing the Square

If the general equation of a circle is given, it can also be converted into standard form using the completing the square method.

For example,

Then,

Completing the square,

Therefore,

(x - 3)² + (y + 2)² = 16

So the centre is (3, -2) and the radius is 4.

👉 Parametric Equations of a Circle

Suppose the origin O(0, 0) is the centre of a circle of radius a, and P(x, y) is a point on the circle. If OP makes an angle θ with the positive x-axis, then from trigonometric ratios,

and

Therefore,

x = a cosθ and y = a sinθ

These are the parametric equations of a circle centred at the origin.

If the centre of the circle is (α, β), the parametric equations are

x = α + a cosθ

y = β + a sinθ

👉 Equation of a Circle When the Endpoints of a Diameter Are Given

If the endpoints of a diameter of a circle are A(x₁, y₁) and B(x₂, y₂), then for any point P(x, y) on the circle, ∠APB = 90°.

Using this condition, the equation of the circle is

(x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0

Example

If the endpoints of a diameter are A(2, 0) and B(6, 0), then

That is,

Therefore,

x² + y² - 8x + 12 = 0

Here, the centre is (4, 0) and the radius is 2.

👉 Quick Formula Revision

👉 Important Points

• In the equation of a circle, the coefficients of x² and y² are equal.
• The general equation of a circle does not contain an xy term.
• From the standard form, the centre and radius can be identified directly.
• From the general form, the centre is (-g, -f) and the radius is √(g² + f² - c).
• The parametric form is especially useful for representing the coordinates of a moving point on a circle.

Once these formulas are clearly understood, it becomes much easier to form equations of circles, determine their centres and radii, and solve many Class 11 Coordinate Geometry problems.