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Transverse and Conjugate Axis of the Hyperbola

We will discuss about the transverse and conjugate axis of the hyperbola along with the examples.


Definition of the transverse axis of the hyperbola:

The transverse axis is the axis of a hyperbola that passes through the two foci.

The straight line joining the vertices A and A’ is called the transverse axis of the hyperbola.

AA' i.e., the line segment joining the vertices of a hyperbola is called its Transverse Axis. The transverse axis of the hyperbola x2a2 - y2b2 = 1 is along the x-axis and its length is 2a.

Transverse and Conjugate Axis of the Hyperbola

The straight line through the centre which is perpendicular to the transverse axis does not meet the hyperbola in real points.

Definition of the conjugate axis of the hyperbola:

If two points B and B' are on the y-axis such that CB = CB’ = b, then the line segment BB’ is called the conjugate axis of the hyperbola. Therefore, the length of conjugate axis = 2b.


Solved examples to find the transverse and conjugate axes of an hyperbola:

1. Find the lengths of transverse and conjugate axis of the hyperbola 16x2 - 9y2 = 144.

Solution:

The given equation of the hyperbola is 16x2 - 9y2 = 144.

The equation of the hyperbola 16x2 - 9y2 = 144 can be written as

x29 - y216 = 1 ……………… (i)

The above equation (i) is of the form x2a2 - y2b2 = 1, where a2 = 9 and b2 = 16.

Therefore, the length of the transverse axis is 2a = 2 ∙ 3 = 6 and the length of the conjugate axis is 2b = 2 ∙ 4 = 8.


2. Find the lengths of transverse and conjugate axis of the hyperbola 16x2 - 9y2 = 144.

Solution:

The given equation of the hyperbola is 3x2 - 6y2 = -18.

The equation of the hyperbola 3x2 - 6y2 = -18 can be written as

x26 - y23 = 1 ……………… (i)

The above equation (i) is of the form x2a2 - y2b2 = -1, where a2 = 6 and b2 = 3.

Therefore, the length of the transverse axis is 2b = 2 ∙ √3 = 2√3 and the length of the conjugate axis is 2a = 2 ∙ √6 = 2√6.

The Hyperbola






11 and 12 Grade Math 

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