Equation of A Circle

What is the Equation of a Circle?

A circle is a closed geometric figure that can be represented on a plane. All the points of a circle are at the same distance from a fixed point called the center of the circle. The point from which all the points of the circle are equidistant is called the center of the circle. The distance between the center and any point on the circle is called its radius. The diameter of the circle is two times its radius. A circle can be represented in a cartesian coordinate system by a general equation which is called the equation of a circle. The equation of a circle can also be generalised in a polar and spherical coordinate system. This section describes the general equation of the circle and how to find the equation of the circle when some data is given about the parts of the circle. 


If a circle is represented in a cartesian plane as shown above, the equation of the circle are given as:


Standard equation: (x - h)2 + (y - k)2 = r2

General equation: x2 + y2 + 2 mx + 2 ny + c = 0 where h = - m, k = n and c = h2 + k2 - r2

Equation of a Circle:

The equation of a circle exists in two different forms namely general form and standard form. The standard equation of a circle with its center of coordinates O(p, q) is written as:


(x - p)2 + (y - q)2 = r2


In the above equation, ‘r’ is the radius of the circle. The abscissa and ordinate of the point that indicates the center of the circle are ‘p’ and ‘q’. 


The general equation of a circle can be obtained by expanding the standard equation of a circle. If the standard equation of a circle as stated above is expanded, we get

x2 + p2 - 2xp + y2 + q2 - 2yq - r2 = 0

x2 + y2 - 2 xp - 2 yq + p2 + q2 - r2 = 0

If in the above equation, we take p = - m, q = -n and p2 + q2 - r2 = k, the equation becomes

x2 + y2 + 2 mx + 2 ny + k = 0

In the above equation, m, n and k are constants. So, the general equation of circle with coordinates of its center as (- m, -n) is written as:


x2 + y2 + 2 mx + 2 ny + k = 0


How to Find the Equation of the Circle with Center at Origin?

When the center of the circle is at the origin, the coordinates of the point indicating the center of the circle is (0, 0). If the circle has a radius = r and a point on the circle with coordinates (x, y), then the equation of the circle is written as follows.

  • Standard Equation of a Circle with Center at Origin

The standard equation of a circle with center at (p , q) is given as:

(x - p)2 + (y - q)2 = r2

When the center is at the origin, the coordinates of the center of the circle is (0, 0). So, substituting the value of p and q equal to 0 we get

(x - 0)2 + (y - 0)2 = r2


x2 + y2 = r2


  • General Equation of Circle with Center at Origin

The general equation of circle with center (-m , -n) is written as 

x2 + y2 + 2 mx + 2 ny + k = 0

If the circle is at the center, the coordinates of the center is (0, 0). So, h = - m = 0 and y = -n = 0. Substituting these values in the above equation, we get 

x2 + y2 + 2 (0) x + 2 (0) y + k = 0

We also know that k = m2 + n2 - r2 = (0)2 + (0)2 - r2 = - r2

x2 + y2 - r2 = 0

Rearranging the terms, we get the equation as: 


x2 + y2 = r2


How to Find the Equation of the Circle in Standard Form From the General Equation?

When the general form of the equation of a circle is given, the completing square method can be used to obtain its standard form. If the general equation of a circle is given as:

x2 + y2 + 2 mx + 2 ny + k = 0


Step 1: Rewrite the equation as:

x2 + y2 - 2 (-m) x - 2 (-n) y + k = 0


Step 2: Add m2 + n2 on both sides of the above equation.

x2 + y2 - 2 (-m) x - 2 (-n) y + k + m2 + n2 = 0 + m2 + n2 


Step 3: Use the algebraic identity for the square of difference between two numbers. 

[Hint: (p - q)2 = p2 + q2 - 2 pq]

[x - (-m)]2 + [y - (-n)]2 + k = m2 + n2 


Step 4: Substitute -m = p, -n = q and k = m2 + n2 - r2

[x - (-m)]2 + [y - (-n)]2 + k = m2 + n2 

[x - p]2 + [y - q]2 + m2 + n2 - r2 = m2 + n2 

[x - p]2 + [y - q]2 - r2 = 0

[x - p]2 + [y - q]2 = r2

Problems on How to Find the Equation of the Circle:

  1. The general equation of the circle is given as x2 + y2 + 4x + 6y - 36 = 0. What is the equation of a circle in standard form? Also determine its radius.

Solution:

The general equation is:

x2 + y2 + 4x + 6y - 36 = 0

It can be rewritten by completing the square as:

x2 + y2 - 2 (-2) x - 2 (-3) y - 36 + (-2)2 + (-3)2 = (-2)2 + (-3)2 

[x - (-2)]2 + [y - (-3)]2 = 36 +13

[x + 2]2 + [y + 3]2 = 49 is the standard equation of the circle.

The radius r = √49 = 7 units


  1. A circle with radius equal to 9 has its center at (5, - 3). How to find the equation of the circle in standard and general form?

Solution:

Given data: (p, q) = (5, -3) and r = 9

Equation of a circle in standard form is 

[x - p]2 + [y - q]2 = r2

[x - 5]2 + [y - (-3)]2 = 92

[x - 5]2 + [y + 3]2 = 81 is the standard equation of the circle.

The general equation is obtained by expanding the above equation with identities as:

x2 + 52 - 2. x. 5 + y2 + 32 + 2. y .3 =81

x2 + y2 - 10x + 6y + 25 + 9 - 81 = 0

x2 + y2 - 10x + 6y - 47 = 0 is the general equation of the circle.


  1. Write the standard form of a circle whose center is on the X axis at a distance of 4 units from the origin and the radius of the circle is 5. What is the equation of a circle in general form obtained from its standard form?

Solution:

Given data: 

Coordinates of the center = (p, q) = (4, 0)

Radius r = 5

Standard form of the equation of a circle is:

[x - p]2 + [y - q]2 = r2

[x - 4]2 + [y - 0]2 = 52

[x - 4]2 + y2 = 25 is the standard equation of the given circle.

General equation of circle is obtained as:

x2 + 42 - 2. x. 4 + y2 = 25

x2 + y2 - 8x + 16 - 25 = 0

x2 + y2 - 8x - 9 = 0 is the general equation of the circle.



Fun Facts About General Equation of Circle:

  • Anaxogoras was the pioneer of the equation of circles. He started his computation of the general form of an equation for a circle in 450 BC.

  • For a circle with its center at the origin, the general and standard form of the equation of a circle are the same.

  • One can also represent the standard and general equation of a circle in complex form.

  • Standard equation of a circle can be more conveniently used to find the radius of a circle.

  • Any line passing through the circle cannot touch the circle at three or more different points. 

FAQ (Frequently Asked Questions)

1. What are the Parts of a Circle?

  • A circle is a closed plane geometric figure which consists of a set of points which are equidistant from a fixed point called the centre of the circle. 

  • The distance between the center and any point on the circle is called its radius.

  • Diameter is a straight line which touches any two points on the circle passing through the center of the circle.

  • The line which touches any two points on the circle is called its chord. Diameter is the longest chord.

  • A portion on the circumference of the circle is called its arc. 

  • The region enclosed by the chord and an arc is called its segment.

  • The region enclosed by an arc and two radii is called the sector of the circle. 

  • The diameter divides the circle into two equal halves which are called the semicircles.

2. What are Tangents and Secants of a Circle?

  • Tangent is a line that touches the circle at only one point. One and only one tangent can be drawn at any point on the circumference of the circle. Tangents drawn to the circle are always perpendicular to the radii of the circle at the point of contact of the tangent to the circle. From a point in the interior of the circle, no tangents can be drawn to the circle. Only two tangents can be drawn to a circle from a point on the exterior of the circle. 

  • Secant is a line that passes through the circle and touches the circle at two different points.