Equation of a Circle – General Form

The general form of the equation of a circle is a more comprehensive representation that allows the circle to be positioned anywhere in the Cartesian plane, not just centered at the origin.

Let’s delve into the details to understand this form and how it relates to the standard form of a circle’s equation.

The general form of the equation of a circle is represented by

x² + y² + 2gx + 2fy + c = 0.

x and y are the arbitrary points on the circumference of the circle, and g, f and c are the constants.

This general form of the equation is used to find the radius and the coordinates of the center of the circle.

The general form of the equation of a circle is

x² + y² + 2gx + 2fy + c = 0.

For all values of g, f and c.

Adding g² + f² on both sides of the equation

we get x² + 2gx + g² + y² + 2fy + f² = g² + f² – c….

(1) (x + g)² = x² + 2gx + g² and

(2) (y + f)² = y² + 2fy + f²

Substituting the values in equation

(1), we have(x + g)² + (y + f)² = g² + f² – c……(2)

Comparing (2) with (x – h)² + (y – k)² = a²,

where (h, k) is the center and a is the radius of the circle.

h = -g, k= -f

a² = g² + f² – c

Therefore, x² + y² + 2gx + 2fy + c = 0, represents the circle with centre (-g, -f) and radius equal to

a² = g² + f² – cIf,

g² + f² > c,

then the radius of the circle is real.

If, g² + f² = c,

then the radius of the circle is zero which shows that the circle is a point that coincides with center.

If, g² + f² < c,

then the radius of the circle become imaginary.

Therefore, it is a circle having a real center and imaginary radius.

Unlike the standard form of the equation of the circle, the general form is difficult to understand and find some meaningful properties of the circle.

Standard Form of a Circle’s Equation

First, recall the standard form of the equation of a circle with center (h, k) and radius r:

(x−h)² + (y−k)²= r²

This equation describes a circle centered at (h, k) with radius r.

General Form of a Circle’s Equation

To convert the standard form to the general form, we expand the equation and rearrange terms. Starting from:

(x−h)² + (y−k)²= r²

Expand the Squares:

(x²−2hx+h²)+(y²−2ky+k²) = r²

Combine Like Terms:

x²+y²−2hx−2ky+h²+k²=r²

Rearrange to General Form:

x²+y²−2hx−2ky+(h²+k²−r²)=0

We can write this as:

x²+y²+Dx+Ey+F=0

where D=−2h E=−2k

F=h²+k²−r²

Components of the General Form

In the general form

x²+y²+ Dx+Ey+F=0:

x²+y² represents the quadratic terms for
x and y.

Dx and Ey are the linear terms. F is the constant term.

Example

Consider the general form:

x²+ y² + 4x -6y -12 = 0

Rewrite Grouping

x²+ 4x + y² -6y = 12

Complete the Square:

(x+2)²−4+(y−3)²−9=12

Simplify:

(x+2)²+(y−3)²=25

Thus, the circle has:

Center: (−2,3) Radius: ✓25=5

Summary

The general form of the equation of a circle,
x²+y²+Dx+Ey+F=0, can be transformed into the standard form to easily identify the center and radius of the circle. This general form is useful as it allows for a more flexible representation of circles in the Cartesian plane.