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Chapter 6: Problem 73

Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form.

Short Answer

Expert verified
The standard form of a hyperbola centered at the origin with a horizontal transverse axis is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(b^2 =c^2 - a^2\).

Step by step solution

01

Identify the Properties

For a hyperbola centered at the origin: The foci are at points (-c, 0) and (c, 0) and the vertices are at (-a, 0) and (a, 0). The distance from each focus to any point (x, y) on the hyperbola will be \(\sqrt{(x \pm c)^2 + y^2}\). Since the distance from the vertex to the focus is the constant difference in distances between a point on the hyperbola and the foci, we know that \(2a = c\).
02

Create the Equation

Using the definition of the hyperbola, the equation relates the distances from the foci to any point (x, y) on the hyperbola is: \(\sqrt{(x - c)^2 + y^2} - \sqrt{(x + c)^2 + y^2} = 2a\). Square both sides to eliminate the square roots and simplify the left-hand side of the equation, remembering that since \(2a = c\), you can substitute c in place of 2a when needed.
03

Simplify the Equation

After the square both sides and simplify, we will obtain the standard form of the hyperbola with a horizontal transverse axis as follows: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(b^2 =c^2 - a^2\).

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