General Form of the Equation of a Circle
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.