Length of Tangent to Circle

Let the equation of the circle be

x² + y² + 2gx + 2fy + c = 0

Let PT be the tangent to the circle from P(x₁, y₁) outside it.

The co-ordinate of the centre C is (− g, − f) and radius r = CT = √(g² + f² − c)

From the right angled triangle PCT,

PT² = PC² − CT²

PT = √(x₁² + y₁² + 2gx₁ + 2fy₁ + c)

If the point P is on the circle, then PT² = 0 (PT is zero).

If the point P is outside the circle then PT² > 0 (PT is real) .

If the point P is inside the circle then PT² < 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.