Tangents from a Point to the Hyperbola

From an external point $P(x_1,y_1)$, force the slope-form tangent $y=mx\pm\sqrt{a^2m^2-b^2}$ through $P$:

Squaring and collecting terms in $m$:

This is the hyperbola's pair-of-tangents quadratic — note the $+b^2$ in the constant term (the one sign difference from the ellipse's analogous quadratic in section 7.2.6, which has $-b^2$). By the roots-and-coefficients relations:

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