Obtain the Cartesian equation of the curve by eliminating the parameter. x=4 t, y= t+2 For the...

Question:

Obtain the Cartesian equation of the curve by eliminating the parameter. x=4 t, y= t+2

For the given polar equation, write an equivalent rectangular equation. {eq}r = -11 csc \ \theta {/eq}

Graph the polar equation. {eq}r = \frac {-1}{2} - sin ? {/eq}

Find {eq}\frac {dy}{dx} \ without \ eliminating \ the \ parameter. x = 1 - 2 \ cos \ t, y = 1 + 6 \ sin t, t \ cannot = to \ n \pi {/eq}

Equations in Non-Cartesian Coordinate Systems:

There are an infinite number of ways to write any one curve. We most often use the Cartesian coordinate system since it is super logical and straightforward. When we are dealing with circles, the super logical straightforward choice is polar coordinates. Other times, neither of these are very convenient, and so we use a parameterization instead. Recall the relations between Cartesian and polar coordinates:

{eq}x = r \cos \theta {/eq}

{eq}y = r \sin \theta {/eq}

{eq}r^2 = x^2+y^2 {/eq}

{eq}\theta = \tan^{-1} \frac{y}{x} {/eq}

{eq}dA = r\ dr\ d\theta {/eq}

Answer and Explanation: 1

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For this first tiny bit, we can write

{eq}\begin{align*} x &= 4t \\ t &= \frac x4 \end{align*} {/eq}

And now we can plug this into {eq}y {/eq}...

See full answer below.


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Evaluating Parametric Equations: Process & Examples

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Chapter 24 / Lesson 3
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Parametric equations are those that define rectangular equations in the context of a single parameter. Use provided examples to evaluate several parametric equations to see how they can be solved, even with unknown variables.


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