Find the equation of the hyperbola satisfying the give conditions: Foci (0, ±13), the conjugate axis is of length 24.
Solution and Explanation
Foci (0, ±13), the conjugate axis is of length 24.
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1. \)
Since the foci are (0, ±13), c = 13.
Since the length of the conjugate axis is 24, 2b = 24
\(⇒\) b = 12.
We know that \(a ^2 + b ^2 = c ^2 . \)
∴ a2 + 122 = 132
\(⇒\) a2 = 169 – 144
= 25
Thus, the equation of the hyperbola is \(\frac{y^2}{25} – \frac{x^2}{144} = 1\)
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Concepts Used:
Hyperbola
Hyperbola is the locus of all the points in a plane such that the difference in their distances from two fixed points in the plane is constant.
Hyperbola is made up of two similar curves that resemble a parabola. Hyperbola has two fixed points which can be shown in the picture, are known as foci or focus. When we join the foci or focus using a line segment then its midpoint gives us centre. Hence, this line segment is known as the transverse axis.
