Equation of a Circle – General Form

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. 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.

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

(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 – c

If, 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.

Leave a Reply

Your email address will not be published. Required fields are marked *