Length of Tangent from an External Point

Length of the tangent drawn from external point (x₁, y₁) to the circle S = 0. Valid only when S₁ > 0 (point outside the circle).

Let the circle $S \equiv x^2 + y^2 + 2gx + 2fy + c = 0$ have centre $C(-g, -f)$ and radius $r = \sqrt{g^2 + f^2 - c}$.

Let $P(x_1, y_1)$ be an external point and $T$ the point of tangency. The radius $CT$ is perpendicular to the tangent $PT$, forming a right angle at $T$.

In right triangle $PCT$:

$$PT^2 = PC^2 - CT^2 = PC^2 - r^2$$

Now, $PC^2 = (x_1 + g)^2 + (y_1 + f)^2 = x_1^2 + 2gx_1 + g^2 + y_1^2 + 2fy_1 + f^2$.

Therefore:

$$PT^2 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + g^2 + f^2 - (g^2 + f^2 - c) = S_1$$ $$PT = \sqrt{S_1}$$

Both tangents from an external point have equal length (since $PT^2 = PT'^2 = S_1$). This is a standard geometric result: tangents from an external point are equal.

The formula requires $S_1 > 0$, confirming $P$ is exterior to the circle.