Tangents from a Point to the Ellipse

From a point $P(x_1,y_1)$ in the plane, using the slope-form tangent $y=mx\pm\sqrt{a^2m^2+b^2}$ from section 7.2.5, and forcing it through $P$:

Squaring both sides to remove the $\pm$ and the square root, then collecting terms in $m$:

This is a quadratic in $m$, so (as for the parabola) it has two roots $m_1,m_2$ in general — confirming that exactly two tangents can be drawn to an ellipse from any external point.

By the standard relations between the roots and coefficients of a quadratic:

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