Tangents from an External Point to a Circle

Tangents from an External Point to a Circle

Setting up. Let $P(x_1,y_1)$ be a point in the plane, lying outside the circle $x^2+y^2=a^2$ (so it is not on the circle itself). Suppose a tangent line from $P$ has slope $m$; by the point-slope form, its equation is

Applying the tangency condition. For this line to actually touch the circle, the perpendicular distance from the centre $O(0,0)$ to it must equal the radius $a$:

Squaring both sides: $(y_1-mx_1)^2=a^2(1+m^2)$. Expanding the left side and collecting every term on one side as a polynomial in $m$:

This is a quadratic equation in $m$ — so it has (in general) two roots, $m_1$ and $m_2$, which are the slopes of the two tangent lines from $P$.

From any point $P$ outside a circle (and in the same plane), exactly two tangents can be drawn to the circle.

Sum and product of the two slopes. Reading the coefficients of the quadratic directly:

$m_1+m_2=\frac{2x_1y_1}{x_1^2-a^2}, \qquad m_1m_2=\frac{y_1^2-a^2}{x_1^2-a^2}$ …