/*! 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 16 Find the center and radius for e... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 16

Find the center and radius for each circle. $$x^{2}+(y-1)^{2}=9$$

Short Answer

Expert verified
The center of the circle is (0, 1) and the radius is 3.

Step by step solution

01

Recognize the Circle Equation Form

Identify the given circle equation and compare it to the standard form of a circle's equation ewline The standard form is: ewline ewline \( \( x - h \)^{2} + ( y - k \)^{2} = r^{2} ewline Here, (h, k) is the center of the circle, and r is its radius.
02

Identify the Center Coordinates

Compare the given equation \( x^{2} + ( y - 1 )^{2} = 9 \) with the standard form. ewline Notice that \( ( x - 0 )^{2} = x^{2} \), so \( h = 0 \).ewline ewline The term \( ( y - 1 )^{2} \) indicates that k = 1.ewline ewline Therefore, the center coordinates are \( (h, k) = (0, 1) \).
03

Determine the Radius

Identify the right side of the equation which represents \( r^{2} = 9 \).ewline ewline To find r, take the square root of both sides: ewline \( r = \sqrt{9} \)ewline \( r = 3 \). ewline Thus, the radius of the circle is 3.

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Key Concepts

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

Standard Form of a Circle
Understanding the standard form of a circle's equation is essential for finding the center and radius of a circle. The standard form looks like this: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewlinenequation should look familiar: ewline ewline ewline Example: ewline ewline ewline helps if you have this form memorized, as you'll see it often!
Center Coordinates
Determine the center of a circle equation by focusing on the changes in the x or y terms: ewline ewline ewline ewline ewline ewline Let鈥檚 look at an example: ewline ewline ewline migration pattern that lead them to be adjusted by the constants provided in the equation ewline ewline ewline ewline For example, in the equation: ewline ewline ewline A clear understanding of this can make many problems easier.
Radius Calculation
One last concept is calculating the radius from your given equation. This also relies on the standard form of the circle equation: ewline ewline ewline ewline ewline ewline Next, note that: ewline ewline ewline ewline This means that your ewline ewline ewline This means: ewline ewline Be careful in distinguishing between r and r^2; each mathematically represent the same character but provide different values dependent on your formula鈥檚 need.

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Most popular questions from this chapter

$$\text {Graph each equation.}$$ $$y=-\sqrt{1-x^{2}}$$

Graph both equations of each system on the same coordinate axes. Solve the system by elimination of variables to find all points of intersection of the graphs. $$\begin{aligned}&(x+1)^{2}+(y-4)^{2}=17\\\&y=x+2\end{aligned}$$

Graph both equations of each system on the same coordinate axes. Use elimination of variables to find all points of intersection. $$ \begin{aligned} &x^{2}+y^{2}=4\\\ &x^{2}-y^{2}=1 \end{aligned} $$

Graph each hyperbola and write the equations of its asymptotes. $$ \frac{y^{2}}{9}-\frac{x^{2}}{16}=1 $$

Graph each relation on a graphing calculator by solving for \(y\) and graphing two functions. $$x=4 y^{2}+4 y+1$$
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