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Chapter 6: Problem 38

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (-2,1),(2,1)\(;\) foci: (-3,1),(3,1)

Short Answer

Expert verified
The standard form of the equation of the hyperbola with the given vertices and foci is \(x^2/4 - (y-1)^2/5 = 1\).

Step by step solution

01

Find the center

The center `(h, k)` of the hyperbola is the midpoint of the line joining the vertices or foci. Calculating the midpoint of two points \((-2, 1)\) and \((2, 1)\), we get \((h, k) = (0, 1)\).
02

Determine the values of a and c

The value of \(a\) is the distance from the center to a vertex, so \(a = 2\). The value of \(c\) is the distance from the center to a focus, so \(c = 3\).
03

Calculate the value of b

The values of \(a\), \(b\) and \(c\) are related by the equation \(c^2 = a^2 + b^2\). Solving this for b, with \(a = 2\) and \(c = 3\), gives \(b^2 = c^2 - a^2 = 9 - 4 = 5\). Thus, \(b = \sqrt{5}\).
04

Write the equation

Use the values found for \(h\), \(k\), \(a\), and \(b\) to write the equation of the hyperbola in standard form. Since this is a horizontal hyperbola, the formula used is \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\). Substituting the values yields \((x-0)^2/2^2 - (y-1)^2/(\sqrt{5})^2 = 1\), simplifying to \(x^2/4 - (y-1)^2/5 = 1\).

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