Before deriving the equation of a circle, let us focus on Circle is a set of all points which are equally spaced from a fixed point in a plane.

- The fixed point is called the centre of the circle.
- The distance between centre and any point on the circumference is called the radius of the circle.

## Equation of a Circle:

### Centre is Origin:

Consider an arbitrary point \(P(x,y)\)

We know that, distance between the point \((x,y)\)

\(\sqrt{x^{2}+y^{2}} = a\)

Therefore, the equation of a circle, with centre as origin is,

\( x^{2}+y^{2} = a^{2}\)

Consider a circle whose centre is at the origin and radius is equal to 8 units.

For the given condition, the equation of a circle is given as

\(x^{2}+y^{2} = 8^{2}\)

### Centre is not origin:

Let \(C(h,k)\)

Therefore radius of a circle is CP.

By using distance formula,

\((x-h)^{2}+(y-k)^{2} = CP^{2}\)

Let radius be a.

Therefore, equation of the circle with centre \((h,k)\)

\((x-h)^{2}+(y-k)^{2} = a^{2}\)

Example: Find the equation of the circle whose centre is \((3,5)\) Solution: Here, centre of the circle is not origin. Therefore, the general equation of the circle is, \((x-3)^2+(y-5)^2 \) \(x^2-6x+9+y^2-10y+25\) \(x^2+y^2-6x-10y+18\) |

Note: The general equation of any type of circle is represented by-

\(x^2~+~y^2~+~2gx~+~2fy~+~c\)

Adding \(g^2~+~f^2\)

\(x^2~+~2gx~+~g^2~+~y^2~+~2fy~+~f^2\)

Since, \((x+g)^2 = x^2+2gx+g^2\)

\((x+g)^2+(y+f)^2 = g^2+f^2-c\)

Comparing (2) with \((x-h)^2+(y-k)^2 = a^2\)

\(h =-g\)

\(a = g^{2}+ f^{2} -c\)

Therefore,

\(x^2+y^2+2gx+2fy+c\)

- If \(g^2+f^2>c\)
, then the radius of the circle is real. - If \(g^2+f^2 = c\)
, then the radius of the circle is zero which tells us that the circle is a point which coincides with the centre. Such type of circle is called as point circle - \(g^2+f^2<c\)
, then the radius of the circle become imaginary. Therefore it is a circle having real centre and imaginary radius.

Example: Equation of a circle is \( x^2+y^2-12x-16y+19 = 0\) Solution: Given equation is of the form \(x^2~+~y^2~+~2gx~+~2fy~+~c\) \(2g\) \(g\) Centre of the circle is \((6,8)\) Radius of the circle = \(√{(-6)^2~+~(-8)^2~-~19}\) |

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