**Circle – General Equation**

**Definition: **A circle is the locus of a point which moves in a plane such that its distance from a fixed point in plane is always a constant. The fixed point is called the center and the constant distance is called the radius of the circle.

**Equation of Circle (h, k) and Radius r:**

The equation of circle is

(x-h)² + (y-k)² = r² …(i)

In particular, if the center is at the origin the equation of circle is

x² + y² = r²

**General Equation of a Circle:**The general equation of the circle

x² + y² + 2gx + 2fy + c = 0 …..(ii)

Where g, f and c are constants

To find the center and radius, (ii) can be written as

(x + g)² + (y + f)² = (√(g² +f²-c) ²

Comparing with the equation of circle given in (i), we get

h = -g, k = -f

and r = √(g² +f²-c)

therefore, the coordinates of the center are (-g, -f) and radius is √(g² +f²-c) , ( since g² +f² ≥ c)

**Example:** Find the equation of the circle with radius 5 whose center lies on the x – axis and passes through the point (2, 3).

**Solution:**Since the radius of the circle is 5 and its center lies on the x- axis,

the equation of the circle is (x – h)² + y² = 25.

It is given that the circle passes through the point (2, 3)

Therefore,

(x – h)² + y² = 25

(2 – h)² + 3² = 25

(2 – h)² = 25 – 9

(2 – h)² = 16

(2-h) = ± 4

If 2-h = 4, then h =-2

If 2-h = -4, then h = 6

Therefore, the equation of circle is

(x + 2)² + y² = 25

(or) (x-6)² + y² = 25.

Hence x² + y² + 4x – 21 = 0

(or)

x² + y² -12x + 11 = 0.