# General Form of the Equation of a Circle

We will discuss about the general form of the equation of a circle.

Prove that the equation x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 always represents a circle whose centre is (-g, -f) and radius = $$\sqrt{g^{2} + f^{2} - c}$$, where g, f and c are three constants

Conversely, a quadratic equation in x and y of the form x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 always represents the equation of a circle.

We know that the equation of the circle having centre at (h, k) and radius = r units is

(x - h)$$^{2}$$ + (y - k)$$^{2}$$ = r$$^{2}$$

x$$^{2}$$ + y$$^{2}$$ - 2hx - 2hy + h$$^{2}$$ + k$$^{2}$$ = r$$^{2}$$

x$$^{2}$$ + y$$^{2}$$ - 2hx - 2hy + h$$^{2}$$ + k$$^{2}$$ - r$$^{2}$$ = 0

Compare the above equation x$$^{2}$$ + y$$^{2}$$ - 2hx - 2hy + h$$^{2}$$ + k$$^{2}$$ - r$$^{2}$$ = 0 with x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 we get, h = -g, k = -f and h$$^{2}$$ + k$$^{2}$$ - r$$^{2}$$ = c

Therefore the equation of any circle can be expressed in the form x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0.

Again, x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0

(x$$^{2}$$ + 2gx + g$$^{2}$$) + (y$$^{2}$$ + 2fy + f$$^{2}$$) = g$$^{2}$$ + f$$^{2}$$ - c

(x + g)$$^{2}$$ + (y + f)$$^{2}$$ = $$(\sqrt{g^{2} + f^{2} - c})^{2}$$

{x - (-g) }$$^{2}$$ + {y - (-f) }$$^{2}$$ = $$(\sqrt{g^{2} + f^{2} - c})^{2}$$

This is of the form (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = r$$^{2}$$ which represents a circle having centre at (- g, -f) and radius $$\sqrt{g^{2} + f^{2} - c}$$.

Hence the given equation x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 represents a circle whose centre is (-g, -f) i.e, (-$$\frac{1}{2}$$ coefficient of x, -$$\frac{1}{2}$$ coefficient of y) and radius = $$\sqrt{g^{2} + f^{2} - c}$$ = $$\sqrt{(\frac{1}{2}\textrm{coefficient of x})^{2} + (\frac{1}{2}\textrm{coefficient of y})^{2} - \textrm{constant term}}$$

Note:

(i) The equation x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 represents a circle of radius = $$\sqrt{g^{2} + f^{2} - c}$$.

(ii) If g$$^{2}$$ + f$$^{2}$$ - c > 0, then the radius of the circle is real and hence the equation x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 represents a real circle.

(iii) If g$$^{2}$$ + f$$^{2}$$ - c = 0 then the radius of the circle becomes zero. In this case, the circle reduces to the point (-g, -f). Such a circle is known as a point circle. In other words, the equation x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 represents a point circle.

(iv) If g$$^{2}$$ + f$$^{2}$$ - c < 0, the radius of the circle $$\sqrt{g^{2} + f^{2} - c}$$ becomes imaginary but the circle is real. Such a circle is called an imaginary circle. In other words, equation x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 does not represent any real circle as it is not possible to draw such a circle.

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