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Chapter 10: Problem 71

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 with a horizontal transverse axis is \(x^2/c^2 - y^2/b^2 = 1\), where 'c' is the distance from the center to a focus and 'b' is the square root of \(c^2 - a^2\), with 'a' being half of the constant difference between distances to foci from any point on the hyperbola.

Step by step solution

01

Define the Foci

Let's define the two foci located at \((-c, 0)\) and \((c, 0)\) on the x-axis at equal distances from the origin, the center of the hyperbola. 'c' is the distance of a focus from the center.
02

Applying the Definition

Take a point P on the hyperbola, with coordinates \((x, y)\). According to the definition of a hyperbola, the difference of the distances PF1 and PF2 (where F1 and F2 are the foci) is a constant, which we will denote as \(2a\), with 'a' being half of the difference. It gives us \(PF1 - PF2 = 2a\), where \(PF1 = \sqrt{(x-c)^2 + y^2}\) and \(PF2 = \sqrt{(x+c)^2 + y^2}\).
03

Simplify the Expression

Squaring both sides will remove the square roots. After simplification, it gives us that \(x^2/c^2 - y^2/c^2 - a^2/c^2 = 1\). Suppose \(b^2 = c^2 - a^2\), we have \(x^2/c^2 - y^2/b^2 = 1\), which is the standard equation of a hyperbola with a horizontal transverse axis.

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