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Chapter 10: Problem 40

Eliminate the parameter and obtain the standard form of the rectangular equation. Hyperbola: \(x=h+a \sec \theta, y=k+b \tan \theta\)

Short Answer

Expert verified
The standard form of the rectangular equation for the hyperbola is \(((x-h)/a)^{2} - ((y-k)/b)^{2} = 1\)

Step by step solution

01

Equations Given

We are given the parametric equations for the hyperbola as \(x=h+a \sec \theta\) and \(y=k+b \tan \theta\)
02

Solve for Secant and Tangent

Let's isolate \(\sec \theta\) and \(\tan \theta\) by subtracting \(h\) and \(k\) from the \(x\) and \(y\) equations respectively. We get \(\sec \theta = (x-h)/a\) and \(\tan \theta = (y-k)/b\)
03

Square Isolated Terms

Now, square both these equations, to get \(\sec^{2} \theta = ((x-h)/a)^{2}\) and \(\tan^{2} \theta = ((y-k)/b)^{2}\)
04

Apply Trigonometric Identity

Recall the trigonometric identity \(\sec^{2} \theta - \tan^{2} \theta = 1\). Substitute from the previous step to get \(((x-h)/a)^{2} - ((y-k)/b)^{2} = 1\)
05

Derive Standard Form

The equation \(((x-h)/a)^{2} - ((y-k)/b)^{2} = 1\) is the standard form of the equation of a hyperbola in rectangular coordinates. Therefore, we have eliminated the parameter \(\theta\) and obtained the standard form of the rectangular equation.

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