Editing Hyperbola (section)

===Tangent===
The simplest way to determine the equation of the tangent at a point <math>(x_0,y_0)</math> is to [[implicit differentiation|implicitly differentiate]] the equation <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1</math> of the hyperbola. Denoting ''dy/dx'' as ''y′'', this produces
<math display="block">\frac{2x}{a^2}-\frac{2yy'}{b^2}= 0 \ \Rightarrow \ y'=\frac{x}{y}\frac{b^2}{a^2}\ \Rightarrow \ y=\frac{x_0}{y_0}\frac{b^2}{a^2}(x-x_0) +y_0.</math>
With respect to <math>\tfrac{x_0^2}{a^2}-\tfrac{y_0^2}{b^2}= 1</math>, the equation of the tangent at point <math>(x_0,y_0)</math> is
<math display="block">\frac{x_0}{a^2}x-\frac{y_0}{b^2}y = 1.</math>

A particular tangent line distinguishes the hyperbola from the other conic sections.<ref>J. W. Downs, ''Practical Conic Sections'', Dover Publ., 2003 (orig. 1993): p. 26.</ref> Let ''f'' be the distance from the vertex ''V'' (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2''f''. The tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°.