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Chapter 6: Problem 52

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
\((x-h)^2/a^2 - (y-k)^2/b^2 = 1\)

Step by step solution

01

Identify the trigonometric identity

Recognize the trigonometric identity \(1 + \tan^2(\theta) = \sec^2(\theta)\), which will help us solve the exercise.
02

Express \(\sec(\theta)\) in terms of x

Isolate \(\sec(\theta)\) in the equation for x, which results in: \(\sec(\theta) = (x-h)/a\)
03

Express \(\tan(\theta)\) in terms of y

Isolate \(\tan(\theta)\) in the equation for y, which results in: \(\tan(\theta) = (y-k)/b\)
04

Eliminate the parameter

Now, square both sides in the derived equations from step 2 and step 3 respectively and then sum them up, which allows us to eliminate the parameter \(\theta\). This leads to: \((x-h)^2/a^2 = \sec^2(\theta)\) and \((y-k)^2/b^2 = \tan^2(\theta)\). Summing these up, we get: \((x-h)^2/a^2 + (y-k)^2/b^2 = 1 + \tan^2(\theta) = \sec^2(\theta)\)
05

Express \tan^2(\theta) and \sec^2(\theta) in terms of x and y

Substitute \(\tan^2(\theta)\) and \(\sec^2(\theta)\) by their corresonding expressions from step 2 and step 3 in the result of step 4. This results in: \((x-h)^2/a^2 + (y-k)^2/b^2 = \sec^2(\theta)\)
06

algebraically manipulate to a standard form

We will finish by rewriting the obtained equation into the standard form of a hyperbola equation: \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\)

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Most popular questions from this chapter

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r^{2}=2 \sin \theta$$

Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{6}{2 \cos \theta-3 \sin \theta}$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{6}{2-3 \sin \theta}$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=-2 \cos \theta$$
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