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Chapter 17: Problem 451

Sketch the locus of \(\mathrm{r}^{2}=\cos \theta\).

Short Answer

Expert verified
The locus of the given polar equation \(r^2 = \cos{\theta}\) can be sketched by converting it to its equivalent Cartesian equation. By doing so, we find that the Cartesian equation is \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\), which represents a circle centered at \((\frac{1}{2}, 0)\) with a radius of \(\frac{1}{2}\). Sketch this circle on the x-y plane to obtain the desired locus.

Step by step solution

01

Identify Cartesian equivalent of the polar equation

By using the relationship \(r^2 = x^2 + y^2\), we can begin to obtain the equivalent Cartesian equation. We are given \(r^2 = \cos{\theta}\). Now we want to express \(\cos{\theta}\) in terms of Cartesian coordinates. Use relationships 1 and 2: 1. \(x = r \cos{\theta}\) (implies that \(\cos{\theta} = \frac{x}{r}\)) 2. \(y = r \sin{\theta}\) Now substitute \(\cos{\theta}\) by \(\frac{x}{r}\) in our equation: \(r^2 = \frac{x}{r}\)
02

Simplify the equation in Cartesian coordinates

Now we have the equation \(r^3 = x\). We need to express it in terms of x and y. Use the relationship \(r^2 = x^2 + y^2\): \(r^3 = (\sqrt{x^2 + y^2})^3\) Now simplify the equation: \(r^3 = x^2 + y^2\) Final Cartesian equivalent is: \(x^2 + y^2 = x\)
03

Identify the shape represented by the Cartesian equation

The Cartesian equivalent of the polar equation is \(x^2 + y^2 = x\). With a bit of manipulation, we can identify the shape of the locus: \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\) This is the equation of a circle in Cartesian coordinates with its center at \((\frac{1}{2}, 0)\) and a radius of \(\frac{1}{2}\).
04

Sketch the locus of the equation

With the information gathered in the previous step, we can now sketch the locus of the polar equation \(r^2 = \cos{\theta}\). 1. Draw the x and y axes (Cartesian coordinates) on the graph. 2. Locate the center of the circle at the point \((\frac{1}{2}, 0)\). 3. From the center, draw a circle with a radius of \(\frac{1}{2}\). The resulting circle is the sketch of the locus of the polar equation given.

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Key Concepts

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

Cartesian Coordinates
Cartesian coordinates are a system that uses two numbers to locate a point on a plane. This system is widely used in both mathematics and science because it makes it easy to plot and visualize functions and equations. The two numbers, typically denoted as \( x \) and \( y \), represent the horizontal and vertical positions on a graph, respectively.
  • The \( x \)-coordinate measures how far to the right or left a point is from the vertical y-axis.
  • The \( y \)-coordinate measures how far up or down a point is from the horizontal x-axis.
Converting equations from polar to Cartesian coordinates involves using relationships between these coordinate systems, such as \( x = r \cos\theta\) and \( y = r \sin\theta\). This conversion helps in visualizing and solving problems involving curves like circles and more complex shapes.
Locus
The term 'locus' in mathematics refers to the set of all points that satisfy a certain condition or equation. Imagine it as a path that a moving point follows under specific rules. In the given exercise, we are trying to find the locus of points described by the equation \( r^2 = \cos\theta\).
  • In polar coordinates, a locus I describes a curve based on the distance from the origin and angle from the positive x-axis.
  • When converting to Cartesian coordinates, the locus is often easier to identify visually, as with the standard 'x' and 'y' axes used.
Understanding the locus is crucial for sketching and analyzing curves, as it essentially tells us where the points on the curve are located in space based on the equations or conditions provided.
Circle
A circle is a perfectly round shape in which every point on its boundary is equidistant from a fixed point called the center. In Cartesian coordinates, the equation of a circle centered at \( (h, k) \) with a radius \( r \) is given by \( (x - h)^2 + (y - k)^2 = r^2 \).
  • The exercise reveals a circle derived from the polar equation \( r^2 = \cos\theta \).
  • Transforming it yields \( x^2 + y^2 = x \), which can be rewritten to show the center and radius more clearly: \( (x - \frac{1}{2})^2 + y^2 = \frac{1}{4} \).
  • This tells us the circle has a center at \( (\frac{1}{2}, 0) \) and a radius of \( \frac{1}{2} \).
Circles are fundamental in geometry, and identifying them through equations is a key skill in both analytical geometry and calculus.

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Most popular questions from this chapter

Draw the locus of the equation: \(\mathrm{r}=2 \cos \theta\).

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