# General Equation of Second Degree Represents a Circle

We will learn how the general equation of second degree represents a circle.

General second degree equation in x and y is

ax$$^{2}$$ + 2hxy + by$$^{2}$$ + 2gx + 2fy + C = 0, where a, h, b, g, f and c are constants.

If a = b(≠ 0 ) and h = 0, then the above equation becomes

ax$$^{2}$$ + ay$$^{2}$$ + 2gx + 2fy + c = 0

x$$^{2}$$ + y$$^{2}$$ + 2 ∙ $$\frac{g}{a}$$ x + 2 ∙ $$\frac{f}{a}$$ y + $$\frac{c}{a}$$ = 0, (Since, a ≠ 0)

x$$^{2}$$ + 2 ∙ x ∙ $$\frac{g}{a}$$ + $$\frac{g^{2}}{a^{2}}$$  + y$$^{2}$$ + 2.y .$$\frac{f}{a}$$ + $$\frac{f^{2}}{a^{2}}$$  = $$\frac{g^{2}}{a^{2}}$$  + $$\frac{f^{2}}{a^{2}}$$  - $$\frac{c}{a}$$

(x + $$\frac{g}{a}$$)$$^{2}$$ + (y + $$\frac{f}{a}$$)$$^{2}$$ = $$(\frac{1}{a}\sqrt{g^{2} + f^{2} - ca})^{2}$$

Which represents the equation of a circle having centre at (-$$\frac{g}{a}$$, -$$\frac{f}{a}$$) and radius = $$\mathrm{\frac{1}{a}\sqrt{g^{2} + f^{2} - ca}}$$

Therefore, the general second degree equation in x and y represents a circle if coefficient of x$$^{2}$$ (i.e., a) = coefficient of y$$^{2}$$ (i.e., b) and coefficient of xy (i.e., h) = 0.

Note: On comparing the general equation x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 of a circle with the general equation of second degree ax$$^{2}$$ + 2hxy + by$$^{2}$$ + 2gx + 2fy + C = 0 we find that it represents a circle if a = b i.e., coefficient of x$$^{2}$$ = coefficient of y$$^{2}$$ and h = 0 i.e., coefficient of xy.

The equation ax$$^{2}$$ + ay$$^{2}$$ + 2gx + 2fy + c = 0, a ≠ 0 also represents a circle.

This equation can be written as

x$$^{2}$$ + y$$^{2}$$ + 2$$\frac{g}{a}$$x + 2$$\frac{f}{a}$$y + $$\frac{c}{a}$$ = 0

The coordinates of the centre are (-$$\frac{g}{a}$$, -$$\frac{f}{a}$$) and radius $$\mathrm{\frac{1}{a}\sqrt{g^{2} + f^{2} - ca}}$$.

Special features of the general equation ax$$^{2}$$ + 2hxy + by$$^{2}$$ + 2gx + 2fy + C = 0 of the circle are:

(i) It is a quadratic equation in both x and y.

(ii) Coefficient of x$$^{2}$$ = Coefficient of y$$^{2}$$. In solving problems it is advisable to keep the coefficient of x$$^{2}$$ and y$$^{2}$$ unity.

(iii) There is no term containing xy i.e., the coefficient of xy is zero.

(iv) It contains three arbitrary constants viz. g, f and c.