**MATHS :: Lecture 08 :: ****Equation of a circle**

#### Circles

** **A circle is defined as the locus of the point, which moves in such a way, that its distance from a fixed point is always constant. The fixed point is called **centre **of the circle and the constant distance is called the **radius** of the circle.

**The equation of the circle when the centre and radius are given**** **

Let C (h,k) be the centre and r be the radius of the circle. Let P(x,y) be any point on the circle.

CP = r CP2 = r2 (x-h)2 + (y-k)2 = r2 is the required equation of the circle.

**Note :**

If the center of the circle is at the origin i.e., C(h,k)=(0,0) then the equation of the circle is x2 + y2 = 2

__The general equation of the circle is x2 + y2 +2gx + 2fy + c = 0 __

Consider the equation x2 + y2 +2gx + 2fy + c = 0. This can be written as

x2 + y2 + 2gx +2fy + g2 + f2 = g2 +f2 – c

(i.e) x2 + 2gx + g2 + y2 +2fy + f2 = g2 +f2 – c

(x + g)2 + (y + f )2 =

+ =

This is of the form (x-h)2+ (y-k)2 = r2

\The considered equation represents a circle with centre (-g,-f) and radius

\ The general equation of the circle is x2 + y2 +2gx + 2fy + c = 0** **

where

c = The Center of the circle whose coordinates are (-g,-f)

r = The radius of the circle =

**Note **

The general second degree equation

ax2 + by2 +2hxy + 2gx + 2fy +c = 0

Represents a circle if

(i) a = b i.e., coefficient of x2 = coefficient of y2

(ii) h = 0 i.e., no xy term

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