Cartesian and parametric equations and asymptotes for a hyperbola

Understanding of Hyperbola Concepts

A hyperbola is a type of curve formed by intersecting a cone with a plane at an angle smaller than the slope of the cone.
Hyperbolae have two disconnected parts called branches, with points on one branch having negative x values and the other positive.
The point of symmetry of each branch is known as the vertex. The line of symmetry through the vertices is the major axis.
The asymptotes of a hyperbola are the lines which the curve approaches but never quite reaches. The hyperbola is symmetric with respect to these axes.

Cartesian and Parametric Equations of a Hyperbola

The standard Cartesian equation for a hyperbola centred at the origin, with its major axis vertical is given by: (y-y_0)^2/a^2 - (x-x_0)^2/b^2 = 1.
If the major axis is horizontal, the equation becomes: (x-x_0)^2/a^2 - (y-y_0)^2/b^2 = 1.
Hyperbolae can also be expressed using parametric equations. The standard parametric equations for a hyperbola centred at the origin with its major axis vertical are x = x_0 + b*sec(theta) and y = y_0 + a*tan(theta).
For major axis horizontal, the parametric equations are x = x_0 + a*sec(theta) and y = y_0 + b*tan(theta).

Asymptotes of a Hyperbola

The asymptotes of a hyperbola are given by the equations y = mx + c, where m is the gradient, and c is the y-intercept.
For a hyperbola with the equation (y-y_0)^2/a^2 - (x-x_0)^2/b^2 = 1, the asymptotes are given by: y = y_0 ± (a/b)(x - x_0).
For a hyperbola with the equation (x-x_0)^2/a^2 - (y-y_0)^2/b^2 = 1, the asymptotes are given by: y = y_0 ± (b/a)(x - x_0).

Practical Tips for Solving Problems

Remember that the vertices of the hyperbola are located at y0±a if the major axis is vertical, and at x0±a if the major axis is horizontal.
When converting from Cartesian to parametric equations, secant and tangent will be in terms of the angle from the origin to the point on the hyperbola.
For asymptote equations, it is helpful to start by drawing a rough sketch of the hyperbola shape to ascertain the position of the axes and the asymptotes.
When calculating the gradient of the asymptotes, remember that it’s ±a/b for a vertical major axis and ±b/a for a horizontal major axis.
In all sections of a hyperbola problem, be mindful of the signs associated with the terms in the equations as they discern the direction and position of the hyperbola.