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 \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 is along the x-axis and its length is 2a.
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 16x\(^{2}\) - 9y\(^{2}\) = 144.
Solution:
The given equation of the hyperbola is 16x\(^{2}\) - 9y\(^{2}\) = 144.
The equation of the hyperbola 16x\(^{2}\) - 9y\(^{2}\) = 144 can be written as
\(\frac{x^{2}}{9}\) - \(\frac{y^{2}}{16}\) = 1 ……………… (i)
The above equation (i) is of the form \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1, where a\(^{2}\) = 9 and b\(^{2}\) = 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 16x\(^{2}\) - 9y\(^{2}\) = 144.
Solution:
The given equation of the hyperbola is 3x\(^{2}\) - 6y\(^{2}\) = -18.
The equation of the hyperbola 3x\(^{2}\) - 6y\(^{2}\) = -18 can be written as
\(\frac{x^{2}}{6}\) - \(\frac{y^{2}}{3}\) = 1 ……………… (i)
The above equation (i) is of the form \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = -1, where a\(^{2}\) = 6 and b\(^{2}\) = 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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