Pair of Tangents from an External Point

Combined equation of the two tangents drawn from external point (x₁, y₁) to circle S = 0, where T = xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c.

Let $P(x_1, y_1)$ be an external point and $Q(h, k)$ be any point on either tangent from $P$ to circle $S = 0$.

The chord of contact (line joining the two contact points) has equation $T_1 = 0$, where $T_1$ is $T$ evaluated at $(x_1, y_1)$.

The key condition: $Q(h, k)$ lies on a tangent from $P$ if and only if $P$, $Q$, and the corresponding contact point are collinear — equivalently, the line $PQ$ is tangent to the circle.

For line $PQ$ to be tangent to $S = 0$, applying the tangent condition (via the discriminant of the system formed by $PQ$ and $S$) leads to:

$$S \cdot S_1 = T^2$$

where:

• $S \equiv x^2 + y^2 + 2gx + 2fy + c$
• $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$
• $T \equiv xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c$

This is the combined (pair) equation of the two tangents from $(x_1, y_1)$. Being a second-degree equation in $x$ and $y$, it represents two lines through $P$.

Note: The angle between the two tangents can be found from this pair equation using the standard formula for the angle between a pair of lines.