An Hyperbola is A Set of All Points on a Plane Difference of whose Distance from 2 Distinct Fixed Points on the Plane is Constant.. The Fixed Points are the 2 Foci of the Hyperbola.
The Distance between the 2 Foci is denoted by letter \(\mathbf{F}\). Half the Distance between the 2 Foci is denoted by letter \(\mathbf{c}\).
The Constant Difference of Distance from 2 Foci is the Length of the Transverse Axis of the Hyperbola and it is denoted by letter \(\mathbf{L}\). Half of Length of the Transverse Axis is called Semi-Transverse Axis and it is denoted by letter \(\mathbf{a}\).
The 2 End Points of the Transverse Axis are the 2 Vertices of the Hyperbola.
The Mid Point of the 2 Vertices (or Mid Point of the 2 Foci) is the Center of the Hyperbola.
The Line Segment Passing through Center of Hyperbola and Perpendicular to it's Transverse Axis is the Conjugate Axis of the Hyperbola.
The 2 End Points of the Conjugate Axis are the 2 Co-Vertices of the Hyperbola. The Distance Between the 2 Co-Vertices is the Length of Conjugate Axis of the Hyperbola and it is denoted by letter \(\mathbf{l}\).
Half of Length of the Conjugate Axis is called Semi-Conjugate Axis and it is denoted by letter \(\mathbf{b}\).
Length of Transverse Axis (\(L)\), Length of Conjugate Axis (\(l)\) and the Distance between the 2 Foci (\(F)\) for any Hyperbola are related as follows
\(F^2=L^2 + l^2\)
Similar relation exists beween the Length of Semi-Transverse Axis (\(a)\), Length of Semi-Conjugate Axis (\(b)\) and Half the Distance between the 2 Foci (\(c)\) given as follows
\(c^2=a^2 + b^2\)
The above 2 equations imply that For any Hyperbola \(L < F\) (hence \(a < c\)).
The Angle that the Transverse Axis of Hyperbola makes with the Poisitive Direction of \(X\)-Axis is the Angular Orientation of the Hyperbola.
The 2 Lines Perpendicular to the Transverse Axis of the Hyperbola at a Distance of \(\frac{L^2}{2F}\) (or \(\frac{a^2}{c}\)) from the Center of Hyperbola on the either side of the Center of Hyperbola are called the 2 Directrices of the Hyperbola.
The Ratio of the Distance Between the 2 Foci To the Length of the Transverse Axis of Hyperbola is called the Eccentricity of the Hyperbola. It is denoted by letter \(e\) and calculated as follows.
\(e=\frac{F}{L}\)
The Eccentricity of an Hyperbola is Always \(> 1\).
The Eccentricity of an Hyperbola can also be given as the Ratio of the Distance Between a Vertex and Center of Hyperbola (which is the Length of Semi-Transverse Axis of Hyperbola) To Distance Between the Directrix and Center of Hyperbola. That is
The Eccentricity of an Hyperbola can also be given as the Ratio of the Distance Between a Focus and a Point on Hyperbola To Distance Between the Directrix (on the same side of Center as the Focus) and that Point on Hyperbola.
The Line Segment Between 2 Points on Hyperbola, Passing through its Focus, Parallel to it's Directrices and Perpendicular to it's Transverse Axis is called the Latus Rectum of Hyperbola.
Each Hyperbola has 2 Latus Recta, one Passing through one of the Focus and the other Passing through the other Focus.
The Length of each Latus Rectum is \(\frac{l^2}{L}\) (or \(\frac{2b^2}{a}\)).
Every Hyperbola has a Pair of Lines Asymptotic to the Hyperbola Passing through the Center of the Hyperbola.
Every Hyperbola has a corresponding Conjugate Hyperbola. A Conjugate Hyperbola is formed by Rotating the Hyperbola by \(90^{\circ}\) (Clockwize or Counter Clockwize) and interchanging Lengths of it's Trasverse Axis and Conjugate Axis.
The Conjugate Hyperbola has the Same Pair of Asymptotic Lines as the Hyperbola.
A Hyperbola having it's Length of Trasverse Axis same as Length of Conjugate Axis is called a Rectangular Hyperbola.
In summary, following are the Parameters / Properties for any given Hyperbola
Length of Transverse Axis/Semi-Transverse Axis
Length of Conjugate Axis/Semi-Conjugate Axis
Length of Latus Rectum
Distance/Half Distance Between 2 Foci
Distance of Directrices from the Center
Eccentricity
Angular Orientation
Coordinates of Center
Coordinates of 2 Vertices
Coordinates of 2 Co-Vertices
Coordinates of 2 Foci
Coordinates of Point of Intersection Between the 2 Directrices and Transverse Axis
Coordinates of Point of Intersection Between the 2 Latus Rectas and Ellipse
Equation of the Transverse Axis
Equation of the Conjugate Axis
Equation of the 2 Directrices
Equation of the 2 Latus Recta
Equation of the 2 Asymptotic Lines
Equation of the Conjugate Hyperbola
Following are 2 types of Hyperbolas
Axis Aligned Hyperbolas: The Axes of these Hyperbolas are Parallel/Perpendicular to the Coordinate Axes. Axis Aligned Hyperbolas can be of the following 2 subtypes
X-Transverse Hyperbola: Hyperbolas having their Transverse Axis Parallel to \(X\)-Axis and Conjugate Axis Parallel to \(Y\)-Axis.
Y-Transverse Hyperbola: Hyperbolas having their Transverse Axis Parallel to \(Y\)-Axis and Conjugate Axis Parallel to \(X\)-Axis.
Non-Axis Aligned Hyperbolas or Rotated Hyperbolas: The Axes of these Hyperbolas are Not Parallel or Perpendicular to Coordinate Axes.
For Axis Aligned Hyperbolas, the General Quadratic Equation in 2 Variables representing the Hyperbola can be given in form of Standard Equations or Implicit Equations.
For Non-Axis Aligned Hyperbolas or Rotated Hyperbolas, the General Quadratic Equation in 2 Variables representing the Hyperbola can be given in form of Implicit Equations only.