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Chapter 5: Problem 57

\(x^2+y^2=9\)

Short Answer

Expert verified
The radius of the circle described by the equation \(x^2 + y^2 = 9\) is 3, and the circle is centered at the origin (0,0).

Step by step solution

01

Understand the form of the equation

The general equation of a circle is given by \(x^2 + y^2 = r^2\), where \(x\) and \(y\) are coordinates on the circle and \(r\) is the radius of the circle. The given equation \(x^2 + y^2 = 9\) falls into this form, where \(r^2 = 9\).
02

Solve for r

To find the radius, \(r\), of the circle, we solve \(r^2 = 9\). Taking the square root of both sides gives us \(r = \sqrt{9}\), which simplifies to \(r = 3\). Therefore, the radius of the circle is 3.
03

Interpreting the result

Our result tells us that the circle described by the equation \(x^2 + y^2 = 9\) has a radius of 3 and is centered at the origin (0,0).

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Key Concepts

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

Circle Radius
The radius of a circle is a crucial component when discussing its geometric properties. Essentially, the radius is the distance from the center of the circle to any point on its circumference.
In mathematical terms, if the circle is centered at the origin of a coordinate system, with its equation represented in the form \[x^2 + y^2 = r^2\],
then the value of the radius is determined by solving \[r = \sqrt{r^2}\].
The current equation, \[x^2 + y^2 = 9\], shows \[r^2 = 9\]. Solving this, the radius \(r\) turns out to be 3. This tells us that each point on the circle is exactly 3 units away from the center located at (0,0).
  • Radius determines the size of the circle.
  • It is measured as the distance from the center to the edge.
Understanding Circle Equations
Circle equations may seem daunting at first, but they actually have a very straightforward structure. The standard form of a circle's equation is \[(x-h)^2 + (y-k)^2 = r^2\],
where \((h, k)\) denotes the center of the circle, and \(r\) is the radius. In simpler terms:
  • If \(h\) and \(k\) are both zero, the circle is centered at the origin.
  • The expression \(x^2 + y^2 = r^2\) specifically indicates a circle at the origin.
The equation tells us everything we need to know about the circle:
  • The center's location.
  • How far out the circle extends, defined by the radius.
For example, the equation \(x^2 + y^2 = 9\) is a simple circle equation. There is no \(h\) or \(k\) affecting the \(x\) and \(y\) terms, meaning the center is right at the origin, \((0, 0)\), with a radius of 3.
Solving Circle Equations
The process of solving circle equations involves a clear understanding and manipulation of its components.
Primarily, solving means determining key attributes like the radius and the center. You often start with equations like \(x^2 + y^2 = 9\), which are already simplified to show their fundamental form.
Follow these steps:
  • Identify \(r^2\): Compare with the general form \(x^2 + y^2 = r^2\). Here, it's clear \(r^2 = 9\).
  • Calculate \(r\): Take the square root of both sides to solve for \(r\). For \(r^2 = 9\), we derive \(r = 3\).
  • Determine the center: If the equation is just \(x^2 + y^2\), it implies the center is at \( (0,0) \).
Understanding each term's role makes it easier to tackle any circle equation. Always check if adjustments are needed, such as shifts in the circle's center.

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Most popular questions from this chapter

\(g(x)=\sqrt{x^2+x}\)

In Exercises 35鈥�46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=-x^3+3 $$

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In Exercises 29 and 30, describe and correct the error in finding the inverse of the function. $$ \begin{aligned} f(x) &=\frac{1}{7} x^2, x \geq 0 \\ y &=\frac{1}{7} x^2 \\ x &=\frac{1}{7} y^2 \\ 7 x &=y^2 \\ \pm \sqrt{7 x} &=y \end{aligned} $$

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