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Chapter 12: Problem 15

Find the center and radius of each circle. Then graph the circle. $$(x-5)^{2}+(y+4)^{2}=49$$

Short Answer

Expert verified
Center: (5,-4), Radius: 7

Step by step solution

01

Identify the Standard Form of the Circle Equation

The equation of a circle in standard form is \( (x-h)^2 + (y-k)^2 = r^2 \). Here, \( h \) and \( k \) represent the coordinates of the center, and \( r \) is the radius.
02

Find the Center of the Circle

Compare the given equation \( (x-5)^2 + (y+4)^2 = 49 \) with the standard form. The center of the circle \( (h,k) \) can be identified from the transformations applied to \( x \) and \( y \). Here, the center \( (h,k) = (5,-4) \).
03

Calculate the Radius

The right side of the equation represents \( r^2 \). Given \( r^2 = 49 \), solve for \( r \) by taking the square root of both sides. Therefore, \( r = \sqrt{49} = 7 \).
04

Plot the Circle

Draw the circle by first plotting the center at \( (5, -4) \). Then, using the radius \( r = 7 \), draw a circle around the center extending 7 units in all directions.

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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 is key to solving problems related to circle equations. The standard form of a circle's equation is given by \((x-h)^2 + (y-k)^2 = r^2\). In this equation:
\h\ represents the x-coordinate of the circle's center,
\k\ represents the y-coordinate of the circle's center,
and \r\ represents the radius of the circle.
Using this form makes it easy to identify the circle's center and radius by simply comparing the given equation. By rewriting any circle equation to this standard form, it becomes straightforward to interpret the main attributes of the circle.
Center of a Circle
Finding the center of a circle involves extracting the coordinates directly from the standard form of the circle's equation. Given the standard equation \((x-h)^2 + (y-k)^2 = r^2\), the center is simply expressed as (h, k).
Let's consider the provided equation: \((x-5)^2 + (y+4)^2 = 49\).
By identifying the transformations applied to \x\ and \y\, we see that the center has been moved to \h = 5\ and \k = -4\.
Therefore, the center of our circle is at point (5, -4). Recognizing the center is crucial for graphing and further analyzing the circle.
Radius Calculation
Calculating the radius of the circle involves analyzing the right-hand side of the circle's standard equation \r^2\. From the given equation \((x-5)^2 + (y+4)^2 = 49\), we see that \r^2 = 49\.
To find the radius \r\, we simply take the square root of both sides:
\[ r = \sqrt{49} = 7 \].
Thus, the radius of the circle is 7 units. Understanding the radius is essential to accurately draw the circle and to understand the extent of the area it covers around its center.

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Most popular questions from this chapter

Graph each hyperbola with center shifted away from the origin. $$ \frac{(x-2)^{2}}{4}-\frac{(y+1)^{2}}{9}=1 $$

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