Find the equation of the hyperbola in the standard form if: length of

1.	Find the equation of the hyperbola in the standard form if: length of the conjugate axis is 3 and the distance between the foci is 5.
Answer» Let the required equation of hyperbola be \(\frac {x^2}{a^2} - \frac {y^2}{b^2} = 1\) Length of conjugate axis = 2b Given, length of conjugate axis = 3 2b = 3 b = 3/2 b² = 9/4 Distance between foci = 2ae Given, distance between foci = 5 2ae = 5 ae = 5/2 a² e² = 25/4 Now, b² = a² (e² – 1) b² = a² e² – a² 9/4 = 25/4 - a² a² = 25/4 - 9/4 ∴ a² = 4 ∴ The required equation of hyperbola is \(\frac {x^2}{4} - \frac {y^2}{\frac {9}{4}} = 1\) i.e., \(\frac {x^2}{4} - \frac {4y^2}{9} = 1\)

Find the equation of the hyperbola in the standard form if: length of the conjugate axis is 3 and the distance between the foci is 5.

Answer»

Let the required equation of hyperbola be \(\frac {x^2}{a^2} - \frac {y^2}{b^2} = 1\)

Length of conjugate axis = 2b

Given, length of conjugate axis = 3

2b = 3

b = 3/2

b² = 9/4

Distance between foci = 2ae

Given, distance between foci = 5

2ae = 5

ae = 5/2

a² e² = 25/4

Now, b² = a² (e² – 1)

b² = a² e² – a²

9/4 = 25/4 - a²

a² = 25/4 - 9/4

∴ a² = 4

∴ The required equation of hyperbola is \(\frac {x^2}{4} - \frac {y^2}{\frac {9}{4}} = 1\)

i.e., \(\frac {x^2}{4} - \frac {4y^2}{9} = 1\)