Let the equation of the circle be

x^{2} + y^{2} + 2gx + 2fy + c = 0

Let PT be the tangent to the circle from P(x_{1}, y_{1}) outside it.

The co-ordinate of the centre C is (− g, − f) and radius r = CT = √(g^{2} + f^{2} − c)

From the right angled triangle PCT,

PT^{2} = PC^{2} − CT^{2}

**PT = √(x _{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c)**

If the point P is on the circle, then PT^{2} = 0 (PT is zero).

If the point P is outside the circle then PT^{2} > 0 (PT is real) .

If the point P is inside the circle then PT^{2} < 0 (PT is imaginary).

The constant c will be positive if the origin is outside the circle, zero if it is on the circle and negative if it is inside the circle.