Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: Equations Given

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

Step 2: 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 \(\tan \theta = (y-k)/b\)

Step 3: Square Isolated Terms

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

Step 4: Apply Trigonometric Identity

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

Step 5: 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.