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Chapter 13: Problem 39

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Short Answer

Expert verified
A circle is defined as all points in a plane that are a constant distance (the radius) from a fixed point (the center). This definition can be written into an equation as: 'The sum of the squares of the horizontal and vertical distances between any point on the circle and the center is constant, and it is the square of the radius'. This is the equation of a circle without using variables.

Step by step solution

01

Understand the Definition of a Circle

A circle is a shape consisting of all points in a plane that are at a consistent distance, called the radius, from a fixed point, known as the center.
02

Translate the Definition to an Equation

Based on the definition, for every point on the circle, the distance from the center to this point is the same. In geometrical terms, this distance is the square root of the sum of the squares of the horizontal distance and the vertical distance. The horizontal and vertical distances are the differences in x and y coordinates, respectively, between a point on the circle and the center.
03

Get rid of the Square Root

To form an equation, it is easier to work with squares as they are easier to compute and compare. We can achieve this by squaring both sides.
04

Summarize the resulting equation

Combining the steps above, we get the following: 'The sum of the squares of the horizontal and vertical distances between any point on the circle and the center is constant, and it is the square of the radius'.

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Most popular questions from this chapter

Graph: \(f(x)=2^{1-x}\). (Section \(12.1,\) Example 4 )

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-2 y^{2}-4 y+1$$

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$x=2(y-1)^{2}+2$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{l}x=2 y^{2}+4 y+5 \\\ (x+1)^{2}+(y-2)^{2}=1\end{array}\right.$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=\frac{1}{2}(y+2)^{2}+1$$
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