Find the equation of the hyperbola satisfying the give conditions: Foci (±5, 0), the transverse axis is of length 8.
Solution and Explanation
Foci (±5, 0), the transverse axis is of length 8.
Here, the foci are on the x-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 (±5, 0), c = 5.
Since the length of the transverse axis is 8,
2a = 8
\(⇒\) a = 4.
We know that
a2 + b2 = c2 .
∴ 42 + b2 = 52
\(⇒\) b2 = 25 – 16= 9
Thus, the equation of the hyperbola is \(\frac{x^2}{16} – \frac{y^2}{9} = 1\)
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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.
