Quadratic equation of Hyperbola

A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates (x, y) of the plane

$A_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0$

provided that the constants A_xx, A_xy, A_yy, B_x, B_y, and C satisfy the determinant condition

$D = \begin{vmatrix} A_{xx} & A_{xy}\\A_{xy} & A_{yy} \end{vmatrix} < 0\,$

A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero

$\Delta := \begin{vmatrix} A_{xx} & A_{xy} & B_{x} \\A_{xy} & A_{yy} & B_{y}\\B_{x} & B_{y} & C \end{vmatrix} = 0$

This determinant Δ is sometimes called the discriminant of the conic section.

Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in Conic section#Eccentricity in terms of parameters of the quadratic form.

The center (x_c, y_c) of the hyperbola may be determined from the formulae

$x_{c} = -\frac{1}{D} \begin{vmatrix} B_{x} & A_{xy} \\B_{y} & A_{yy} \end{vmatrix}$

$y_{c} = -\frac{1}{D} \begin{vmatrix} A_{xx} & B_{x} \\A_{xy} & B_{y} \end{vmatrix}$

In terms of new coordinates, ξ = x − x_c and η = y − y_c, the defining equation of the hyperbola can be written

$A_{xx} \xi^{2} + 2A_{xy} \xi\eta + A_{yy} \eta^{2} + \frac{\Delta}{D} = 0$

The principal axes of the hyperbola make an angle Φ with the positive x-axis that equals

$\tan 2\Phi = \frac{2A_{xy}}{A_{xx} - A_{yy}}$

Rotating the coordinate axes so that the x-axis is aligned with the transverse axis brings the equation into its canonical form

$\frac{{x}^{2}}{a^{2}} - \frac{{y}^{2}}{b^{2}} = 1$

The major and minor semiaxes a and b are defined by the equations

$a^{2} = -\frac{\Delta}{\lambda_{1}D} = -\frac{\Delta}{\lambda_{1}^{2}\lambda_{2}}$

$b^{2} = -\frac{\Delta}{\lambda_{2}D} = -\frac{\Delta}{\lambda_{1}\lambda_{2}^{2}}$

where λ₁ and λ₂ are the roots of the quadratic equation

$\lambda^{2} - \left( A_{xx} + A_{yy} \right)\lambda + D = 0$

For comparison, the corresponding equation for a degenerate hyperbola is

$\frac{{x}^{2}}{a^{2}} - \frac{{y}^{2}}{b^{2}} = 0$

The tangent line to a given point (x₀, y₀) on the hyperbola is defined by the equation

$E x + F y + G = 0$

where E, F and G are defined

$E = A_{xx} x_{0} + A_{xy} y_{0} + B_{x}$

$F = A_{xy} x_{0} + A_{yy} y_{0} + B_{y}$

$G = B_{x} x_{0} + B_{y} y_{0} + C$

The normal line to the hyperbola at the same point is given by the equation

$F \left( x - x_{0} \right) - E \left( y - y_{0} \right) = 0$

The normal line is perpendicular to the tangent line, and both pass through the same point (x₀, y₀).

From the equation

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \qquad 0 < b \leq a$

the basic property that with $r_1 \,\!$ and $r_2 \,\!$ being the distances from a point $(x,y) \,\!$ to the left focus $(-a e , 0) \,\!$ and the right focus $(a e , 0) \,\!$ one has for a point on the right branch that