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Chapter 10: Problem 15

Identify the curve by finding a Cartesian equation for the curve. \(r^{2}=5\)

Short Answer

Expert verified
The curve is a circle centered at the origin with radius \( \\sqrt{5} \).

Step by step solution

01

Understand the Equation

The equation given is \( r^{2} = 5 \), where \( r \) is the radial coordinate in polar coordinates. This represents all points at a fixed distance from the origin.
02

Recall Cartesian to Polar Conversion

In polar coordinates, the relationship between \( r \) and Cartesian coordinates \( (x, y) \) is given by \( r = \sqrt{x^2 + y^2} \). This is crucial for converting the given polar equation to Cartesian form.
03

Substitute the Polar Equation

Substitute \( r = \sqrt{x^2 + y^2} \) into the given polar equation \( r^2 = 5 \). This yields \( (\sqrt{x^2 + y^2})^2 = 5 \).
04

Simplify the Equation

Once substituted, simplify the equation. The equation \( (\sqrt{x^2 + y^2})^2 = 5 \) becomes \( x^2 + y^2 = 5 \).
05

Identify the Cartesian Equation Type

The resultant equation \( x^2 + y^2 = 5 \) is recognized as the equation of a circle in Cartesian coordinates, with the circle centered at the origin (0, 0) and with a radius \( \sqrt{5} \).

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Key Concepts

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

Polar Coordinates
Polar coordinates provide a way of describing points in a plane using angles and distances. Instead of horizontal and vertical distances like Cartesian coordinates, polar coordinates use:
  • A radial coordinate, \( r \), which measures how far the point is from the origin.
  • An angular coordinate, \( \theta \), which measures the angle made with the positive x-axis.
This system is particularly useful for problems involving curves that naturally "wrap around" a center point, such as circles and spirals. When given an equation in polar coordinates, like \( r^2 = 5 \), the challenge often is to convert it into the more familiar Cartesian coordinate system to understand its shape and location better. Essentially, polar coordinates offer a different perspective, especially useful when dealing with circular or rotational symmetry.
Cartesian Coordinates
The Cartesian coordinate system is perhaps the most familiar method of plotting points on a plane. It uses two perpendicular axes: the horizontal x-axis and the vertical y-axis.
  • Every point on the plane is defined by a pair of numbers \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position.
  • The system is excellent for describing straight lines, parabolas, and other algebraic curves directly.
To convert from polar coordinates \((r, \theta)\) to Cartesian coordinates, the formulas used are:
  • x = r \cos(\theta)
  • y = r \sin(\theta)
This allows any point described in polar terms to be expressed in the Cartesian system, aligning with our typical understanding of geometry. In our exercise solution, identifying the circle's equation \(x^2 + y^2 = 5\) involved transforming its representation from polar to Cartesian to easily recognize the geometry involved.
Equation of a Circle
The equation of a circle in the Cartesian coordinate system is typically expressed as \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \((h, k)\) represents the circle's center, and \(r\) is the radius. When a circle is centered at the origin, the equation simplifies to \(x^2 + y^2 = r^2\), which is our case where \(x^2 + y^2 = 5\).
This standard form helps easily visualize and understand the circle's size and position on the plane:
  • The center is at the origin \((0,0)\).
  • The radius is \( \sqrt{5} \) since we have \(r^2 = 5\), indicating the circle stretches out evenly from the center by that distance.
Recognizing and converting these equations across coordinate systems allows us to handle geometrical problems more effectively, especially when transitioning from abstract formulations to practical interpretations and graphical representations.

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Most popular questions from this chapter

\(45-48\) Find the exact length of the polar curve. $$ r=2 \cos \theta, \quad 0 \leqslant \theta \leqslant \pi $$

Find the area of the region that lies inside the first curve and outside the second curve. $$ r=3 \sin \theta, \quad r=2-\sin \theta $$

Identify the type of conic section whose equation is given and find the vertices and foci. $$ x^{2}-2 x+2 y^{2}-8 y+7=0 $$

Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r=1+2 \sin (\theta / 2) \quad\) (nephroid of Freeth)

Write a polar equation of a conic with the focus at the origin and the given data. Parabola, directrix \(x=-3\)
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