Find the equation of the hyperbola satisfying the give conditions: Vertices (±7, 0), \(e = \frac{4}{3}\)
Solution and Explanation
Vertices (±7, 0), \(e =\frac{ 4}{3} \)
Here, the vertices 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 vertices are (±7, 0), a = 7.
It is given that \(e =\frac{ 4}{3}\)
\(∴ \frac{c}{a} = \frac{4}{3} [e=\frac{c}{a}]\)
\(⇒ 3c = 4a\)
\(⇒ 3c = 4(7)\)
\(⇒ c = \frac{28}{3}\)
We know that \( a^2 + b^2 = c^2\)
\(7^2 + b^2 = (\frac{28}{3})^2\)
\(b^2 = \frac{784}{9} – 49\)
\(=\frac{ (784 – 441)}{9}\)
\(= \frac{343}{9}\)
Thus, the equation of the hyperbola is \(\frac{x^2}{49} – \frac{9y^2}{343} = 1\)
Top Questions on Hyperbola
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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.
