A hyperbola is the locus of a point in a plane such that the difference of its distances from two fixed points is a constant. A conic section whose eccentricity is greater than
Standard Equation
Let the two fixed points (called foci) be
By using distance formula,
Since the difference of two sides of a triangle is less than the third, we have
Let
This is the equation of the hyperbola in standard form.
Graph
The graph of a hyperbola consists of two branches that go to infinity. The two straight lines
The vertices of the hyperbola are
The eccentricity
Since
Conjugate Hyperbola
The hyperbola whose transverse axis is along the y-axis and the conjugate axis along the x-axis is called the conjugate hyperbola. Its equation is
Its vertices are
The coordinates of the foci are
Hyperbola whose Asymptotes are the Axes of Coordinates
If the foci of a hyperbola be at
The asymptotes of such hyperbola are the axes of coordinates as shown in the above graph.
Equation of the Hyperbola centered at
The equation of a hyperbola whose centre is at
Its vertex is
The equation of directrix is
Similarly, we can evaluate the hyperbola given by the equation
Its vertex is
The equation of directrix is
Previous: Ellipse