We will learn how to find the equation of a circle whose centre and radius are given.

**Case I:** If the centre and radius of a circle be given, we
can determine its equation:

To find the equation of the circle whose centre is at the origin O and radius r units:

Let M (x, y) be any point on the circumference of the required circle.

Therefore, the locus of the moving point M = OM = radius of the circle = r

⇒ OM\(^{2}\) = r\(^{2}\)

⇒ x\(^{2}\) + y\(^{2}\) = r\(^{2}\), which is the required equation of the circle.

**Case II:** To find the equation of the circle whose centre is
at C (h, k) and radius r units:

Let M (x, y) be any point on the circumference of the requited circle. Therefore, the locus of the moving point M = CM = radius of the circle = r

⇒ CM\(^{2}\) = r\(^{2}\)

⇒ (x - h)\(^{2}\) + (y - k)\(^{2}\) = r\(^{2}\), which is the required equation of the circle.

**Note: **

(i) The above equation is known as the central from of the equation of a circle.

(ii) Referred to O as pole and OX as initial line of polar co-ordinate system, if the polar co-ordinates of M be (r, θ) then we shall have,

r = OM = radius of the circle = a and ∠MOX = θ.

Then, from the above figure we get,

x = ON = a cos θ and y = MN = a sin θ

Here, x = a cos θ and y = a sin θ represent the parametric equations of the circle x\(^{2}\) + y\(^{2}\) = r\(^{2}\).

Solved examples to find the equation of a circle:

**1.** Find the equation of a circle whose centre is (4, 7) and
radius 5.

**Solution:**

The equation of the required circle is

(x - 4)\(^{2}\) + (y - 7)\(^{2}\) = 5\(^{2}\)

⇒ x\(^{2}\) - 16x + 16 + y\(^{2}\) - 14y + 49 = 25

⇒ x\(^{2}\) + y\(^{2}\) - 16x - 14y + 40 = 0

**2.** Find the equation of a circle whose radius is 13 and the
centre is at the origin.

**Solution:**

The equation of the required circle is

x\(^{2}\) + y\(^{2}\) = 13\(^{2}\)

⇒ x\(^{2}\) + y\(^{2}\) = 169

**●** **The Circle**

**Definition of Circle****Equation of a Circle****General Form of the Equation of a Circle****General Equation of Second Degree Represents a Circle****Centre of the Circle Coincides with the Origin****Circle Passes through the Origin****Circle Touches x-axis****Circle Touches y-axis****Circle Touches both x-axis and y-axis****Centre of the Circle on x-axis****Centre of the Circle on y-axis****Circle Passes through the Origin and Centre Lies on x-axis****Circle Passes through the Origin and Centre Lies on y-axis****Equation of a Circle when Line Segment Joining Two Given Points is a Diameter****Equations of Concentric Circles****Circle Passing Through Three Given Points****Circle Through the Intersection of Two Circles****Equation of the Common Chord of Two Circles****Position of a Point with Respect to a Circle****Intercepts on the Axes made by a Circle****Circle Formulae****Problems on Circle**

**11 and 12 Grade Math**__From Equation of a Circle____ to HOME PAGE__

**Didn't find what you were looking for? Or want to know more information
about Math Only Math.
Use this Google Search to find what you need.**

## New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.