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.

● The Circle

• Definition of Circle
• Equation of a Circle
• General Form of the Equation of a Circle
• General Equation of Second Degree Represents a Circle
• Centre of the Circle Coincides with the Origin
• Circle Passes through the Origin
• Circle Touches x-axis
• Circle Touches y-axis
• Circle Touches both x-axis and y-axis
• Centre of the Circle on x-axis
• Centre of the Circle on y-axis
• Circle Passes through the Origin and Centre Lies on x-axis
• Circle Passes through the Origin and Centre Lies on y-axis
• Equation of a Circle when Line Segment Joining Two Given Points is a Diameter
• Equations of Concentric Circles
  
• Circle Passing Through Three Given Points
• Circle Through the Intersection of Two Circles
• Equation of the Common Chord of Two Circles
• Position of a Point with Respect to a Circle
• Intercepts on the Axes made by a Circle
• Circle Formulae
• Problems on Circle

11 and 12 Grade Math 

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