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Chapter 6: Problem 39

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: \((\pm 1,0) ;\) asymptotes: \(y=\pm 5 x\)

Short Answer

Expert verified
The standard form of the equation of the hyperbola is \(x^2 - \frac{y^2}{25} = 1\)

Step by step solution

01

Identify the values of 'a' and 'b'

From the vertices \((卤1,0)\), 'a' can be identified as 1. From the asymptotes \(y=卤5x\), identify 'b' as 5 because the slopes of the asymptotes 卤5 indicate that the ratio b/a is 5/1. Therefore, 'b' equals 5.
02

Write the standard form of the equation

Now that the values for 'a' and 'b' are known, the standard form equation for the hyperbola can be obtained by substituting 'a' by 1 and 'b' by 5 into the formula \((\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1)\). Therefore, the equation becomes \((\frac{x^2}{1^2} - \frac{y^2}{5^2} = 1)\) which simplifies to \(x^2 - \frac{y^2}{25} = 1\).

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