/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Write the equation in rectangula... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 59

Write the equation in rectangular coordinates and identify the curve. $$r=\frac{4}{2-\cos \theta}$$

Short Answer

Expert verified
The Cartesian equation of the given polar equation \(r=\frac{4}{2-\cos\theta}\) is \(x^3 - 3x^2 + 2xy^2 - y^2 + y^4 = 0\). After graphing this equation, it is found to be a lima莽on curve.

Step by step solution

01

Convert the polar equation to Cartesian form using conversion formulas

Begin by writing down the conversion formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\). Now, multiply both sides of the polar equation by \(r\): \(r^2 = 4r \div (2 - \cos\theta)\) Substitute the conversion formulas into the equation, replacing \(r^2\) with \(x^2 + y^2\) and \(r\) with \(x\div\cos\theta\): \(x^2 + y^2 = 4\left(\frac{x}{\cos\theta} \div (2 - \cos\theta)\right)\)
02

Simplify the equation further

The current equation is still not solved for x and y. Explore methods to simplify the equation further. \(x^2 + y^2 = 4\left(\frac{x}{\cos\theta} \div (2 - \cos\theta)\right)\) Expand the denominator by multiplying both sides by \((2 - \cos\theta)\): \((x^2 + y^2)(2 - \cos\theta) = 4x\) Rewrite the term \(\cos\theta\) as \(x \div r\), and use the conversion formula \(r = \sqrt{x^2 + y^2}\): \((x^2 + y^2)\left(2 - \frac{x}{\sqrt{x^2 +y^2}}\right) = 4x\) Multiply both sides by \(\sqrt{x^2 +y^2}\): \((x^2 + y^2)(2\sqrt{x^2 + y^2} - x) = 4x\sqrt{x^2 +y^2}\)
03

Identify the curve

Now that we've rewritten the equation in terms of x and y, we can attempt to recognize the resulting equation as a particular type of curve. \((x^2 + y^2)(2\sqrt{x^2 + y^2} - x) = 4x\sqrt{x^2 +y^2}\) At a glance, we can see that this is a conic section equation. We can further simplify it to make it more identifiable. Move all terms to one side: \(x^3 - 3x^2 + 2xy^2 - y^2 + y^4 = 0\) From looking at this equation, we can notice a pattern. \(x^2(x - 3) + y^2(2x - 1) + y^4 = 0\) It still isn't clear what type of curve this is, and it could be challenging to continue simplifying. However, a common approach to identifying curves is to use a graphing calculator or plotting software to visualize the equation. When graphing this equation, we can see that it forms a lima莽on curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conic Sections
Conic sections are curves that are formed by the intersection of a plane and a cone. Understanding different kinds of conic sections helps us analyze various geometrical shapes.
These shapes are defined by their unique equations and include:
  • Circle: The set of points equidistant from a center point.
  • Ellipse: An elongated circle or oval shape.
  • Parabola: A mirror-symmetrical, U-shaped curve.
  • Hyperbola: Consists of two separate curves pointing away from each other.
Conic sections are very useful in fields like astronomy, engineering, and physics. They help model various phenomena, planetary orbits being one such example. When converting polar equations to Cartesian coordinates, as seen in the exercise, we often encounter conic sections, which we can then analyze using familiar algebraic tools.
Lima莽on Curve
A Lima莽on curve is a fascinating type of polar curve. It's defined by polar equations with forms like \(r = a + b \, \cos\theta\) or \(r = a + b \, \sin\theta\). These curves can exhibit interesting shapes depending on the values of \(a\) and \(b\), including:
  • A cardioid when \(a = b\).
  • An inner loop when \(|a| < |b|\).
  • A dimpled curve when \(|a| > |b|\), but \(a + b eq 0\).
  • A convex shape when \(a = 0\).
Understanding the properties of Lima莽on curves is important in studying complex shapes in mathematics and physics. As we saw in the exercise, sometimes equations can be converted and recognized as a specific type of Lima莽on curve through graphing or mathematical insight.
Cartesian Conversion
Cartesian conversion involves changing an equation from polar coordinates to Cartesian coordinates. Polar coordinates are often given in the form \(r = f(\theta)\), while Cartesian coordinates are expressed as \(x\) and \(y\).
To interface between these systems, we use specific conversion formulas:
  • \(x = r \cos\theta\)
  • \(y = r \sin\theta\)
  • \(r^2 = x^2 + y^2\)
In the exercise, using these formulas allowed us to change the polar equation into a more familiar Cartesian form, which makes identifying and analyzing the curve easier. Such conversions are essential in helping us work across different coordinate systems and apply various mathematical tools for problem-solving.
Polar Coordinates
Polar coordinates are a two-dimensional coordinate system where each point is determined by a distance and an angle. Unlike Cartesian coordinates that use \(x\) and \(y\), polar coordinates use:
  • \(r\): the radial distance from the origin to the point.
  • \(\theta\): the angle measured from the positive x-axis to the point.
This system is especially useful for circular and irregular radial patterns. For instance, polar coordinates simplify equations involving circles and spirals, which are more complex in Cartesian form.
In converting equations, as in our exercise, polar coordinates provide a different perspective and facilitate understanding of rotational and radial dynamics. They are widely used in fields like engineering, navigation, and physics to describe systems involving angles and distances from a central point.

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Most popular questions from this chapter

Let \(l\) be a line and let \(F\) be a point not on \(l .\) You have seen that the set of points \(P\) for which $$ d(F, P)=d(l, P) $$ is a parabola. Show that, if \(0

Express the curve by an equation in \(x\) and \(y\) then sketch the curve. $$x(t)=\sec t, \quad y(t)=\tan t \quad 0 \leq t \leq \frac{1}{4} \pi$$

Find the length of the polar curve. $$r=e^{2 \theta} \quad \text { from } \theta=0 \text { to } \theta=2 \pi$$

Find the points \((x, y)\) at which the curve has: (a) a horizontal tangent: (b) a vertical tangent. Then sketch the curve. $$x(t)=t^{2}-2 t, \quad y(t)=t^{3}-3 t^{2}+2 t$$

Find a parametrization $$x \quad x(t), \quad y \quad y(t)\quad t \in[0,1]$$ for the given curve. The line segment from (3,7) to (8,5)
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