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

Definition of the transverse axis of the hyperbola:

The across Axis is the axis of a hyperbola that passes through the two focal points.

The straight line connecting the corners A and A ‘is called across Axis of the Hyperbole.

AA ‘, ie the line segment that connects the corner points of a hyperbola, is called the 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 center point that is perpendicular to the across Axis does not meet the hyperbola in real points.

Definition of the conjugate axis of the hyperbola:

If two points B and B ‘lie on the y-axis with CB = CB’ = b, then the line segment is called BB ‘ conjugate axis of the hyperbola. Hence the conjugate axis length = 2b.

Solved examples to find that transverse and conjugate axes a 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 ……………… (me)

The above equation (i) has 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 ……………… (me)

The above equation (i) has 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 ****hyperbole**

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