Hyperbola Formula

A hyperbola is the set of all the points in a plane. The difference between these distances from two fixed points in the plane will be constant. Here difference means the distance to the farther point minus the distance to the closest point. The two fixed points will be the foci and the mid-point of the line segment joining the foci will be the center of the hyperbola. In geometrical mathematics, Hyperbola is an interesting topic. Here we will discuss the Hyperbola formula with examples. Let us learn the concept!

Hyperbola Formula

What is Hyperbola?

In simple words, hyperbola looks similar to the mirrored parabolas. The two halves are referred to as the branches. When the plane intersects on the halves of a right circular cone angle of which will be parallel to the axis of the cone, then hyperbola is formed.

A hyperbola contains two foci and two vertices. The foci of the hyperbola are away from its center and vertices. The line through the foci is the transverse axis. Also, the line through the center and perpendicular to the transverse axis is known as the conjugate axis. The points at which the hyperbola intersects the transverse axis are known as the vertices of the hyperbola.

General Equation of the hyperbola is:\(\large \frac{(x-x_{0})^{2}}{a^{2}}-\frac{(y-y_{0})^{2}}{b^{2}}=1\)

\(x_{0}, y_{0}\)  are the center points, a is a semi-major axis and b is a semi-minor axis.

• The distance between the two foci will always be 2c
• The distance between two vertices will always be 2a. This is also the length of the transverse axis.
• The length of the conjugate axis will be 2b. Here \(b = \sqrt{(c^2 – a^2)}\)

Source: en.wikipedia.org

Some Basic Formula for Hyperbola

• Major Axis: The line that passes through the center, the focus of the hyperbola and vertices is the Major Axis. Length of the major axis = 2a. The equation is: \(\large y=y_{0}\)
• Minor Axis: The line perpendicular to the major axis and passes by the middle of the hyperbola are the Minor Axis. Length of the minor axis = 2b. The equation is: \(\large x=x_{0}\)
• Eccentricity: The variation in the conic section being completely circular is eccentricity. It is usually greater than 1 for hyperbola. Eccentricity is \(2\sqrt{2}\) for a regular hyperbola. The formula for eccentricity is: \(\large \frac{\sqrt{a^{2}+b^{2}}}{a}\)
• Asymptotes: Two bisecting lines that are passing by the center of the hyperbola that doesn’t touch the curve is known as the Asymptotes. The equation is:
  • \(\large y=y_{0}+\frac{b}{a}x-\frac{b}{a}x_{0}\)
  • \(\large y=y_{0}-\frac{b}{a}x+\frac{b}{a}x_{0}\)
• Directrix of a hyperbola: Directrix of a hyperbola is a line that is used for generating the curve. We can define it as the line from which the hyperbola curves away. This is perpendicular to the axis of symmetry. Its equation is: \(\large x=\frac{\pm a^{2}}{\sqrt{a^{2}+b^{2}}}\)
• Vertex: The point of the branch that is stretched and is closest to the center is the vertex. The vertex points are \((a, y_0) and (-a, y_0)\)
• Focus (foci): On a hyperbola, both focus is the fixed points in such a way that the difference between the distances is always constant. The two focal points are
  • \((\large\left(x_{0}+\sqrt{a^{2}+b^{2}},y_{0}\right)\)\)
  • \(\large \left(x_{0}-\sqrt{a^{2}+b^{2}},y_{0}\right)\)

Solved Examples for Hyperbola Formula

Q.1: The equation of the hyperbola is:\(\frac{(x-4)^{2}}{9^{2}}-\frac{(y-2)^{2}}{7^{2}}\)

Find the following measures in it, Vertex, Asymptote, Major Axis, Minor Axis, and Directrix?

Solution: Given,

• \(x_{0}=4\)
• \(y_{0}=2\)
• a = 9
• b = 7

The vertex point: (a,y0) and (−a,y0) are (9,2) and (−9,2)

Asymptote: \(y=2+\frac{7}{9}x-\frac{37}{9} =2-\frac{7}{9}x-\frac{23}{9}\)

y=2+0.77x+4.1=6.1+0.77x

y=2-0.77x+2.5=4.5+0.77x

Major Axis: \(y=y_{o}  and y_{o}=2\)

Minor Axis: \(x=x_{o}  and  x_{o} =4\)

Directrix: \(x=\frac{\pm 9^{2}}{\sqrt{9^{2}+7^{2}}} = \pm \frac{81}{\sqrt{81+49}}=7.1\)