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Chapter 2: Problem 20

Write the standard form of the equation and the general form of the equation of each circle of radius \(r\) and center \((h, k)\). Graph each circle. $$ r=7 ;(h, k)=(-5,-2) $$

Short Answer

Expert verified
Standard form: \[ (x + 5)^2 + (y + 2)^2 = 49 \]. General form: \[ x^2 + y^2 + 10x + 4y - 20 = 0 \].

Step by step solution

01

Write the standard form of the equation of the circle

The standard form of the equation of a circle with radius \( r \) and center \((h, k)\) is given by \[ (x - h)^2 + (y - k)^2 = r^2 \]. Plug in the given values: \( r = 7 \), \( h = -5 \), and \( k = -2 \). Thus, the standard form is: \[ (x + 5)^2 + (y + 2)^2 = 49 \].
02

Expand the standard form to get the general form

To obtain the general form, expand the equation from Step 1. First, expand \( (x + 5)^2 \): \[ (x + 5)^2 = x^2 + 10x + 25 \]. Next, expand \( (y + 2)^2 \): \[ (y + 2)^2 = y^2 + 4y + 4 \]. Combine these expansions and set equal to 49: \[ x^2 + 10x + 25 + y^2 + 4y + 4 = 49 \].
03

Simplify to get the general form of the equation

Simplify the equation by combining like terms and moving all terms to one side: \[ x^2 + y^2 + 10x + 4y + 29 - 49 = 0 \]. Finally, this simplifies to: \[ x^2 + y^2 + 10x + 4y - 20 = 0 \]. This is the general form of the equation of the circle.
04

Graph the circle

To graph the circle, plot the center at \((-5, -2)\) on the coordinate plane. From there, mark points that are 7 units away from the center in all directions. Connect these points smoothly to form the circle, ensuring the radius is consistent throughout.

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Key Concepts

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

Standard Form
The standard form of the equation of a circle is a very useful way to understand and work with circles.
It's given by \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( h \) and \( k \) are the coordinates of the center, and \( r \) is the radius.
This form is great because it immediately shows you the most important properties of the circle: its center and radius.
For our example with \( r = 7 \) and center at \( (-5, -2) \), plugging these values into the standard form equation, we get: \[ (x + 5)^2 + (y + 2)^2 = 49 \] This tells us directly that the circle has a radius of 7 and is centered at (-5, -2).
Understanding this form makes it much easier to graph and visualize the circle.
General Form
The general form of the equation of a circle looks different from the standard form but represents the same circle.
It's written as \[ Ax^2 + Ay^2 + Dx + Ey + F = 0 \] To turn the standard form into the general form, we need to expand and simplify.
From our example, the standard form \[ (x + 5)^2 + (y + 2)^2 = 49 \] can be expanded to: \[ x^2 + 10x + 25 + y^2 + 4y + 4 = 49 \] Combining like terms and moving everything to one side gives us the general form: \[ x^2 + y^2 + 10x + 4y - 20 = 0 \] The general form is longer, but it is useful in algebra for solving systems of equations involving circles.
Circle Graphing
Graphing a circle is straightforward once you have its equation, especially in the standard form.
First, plot the center of the circle on the coordinate plane. For our example, that's the point \( (-5, -2) \).
Next, use the radius to mark points around the center. Since the radius is 7, you mark points that are 7 units away in all directions.
These points could be directly above, below, to the left, and right of the center, or in between these directions.
Once you have a set of points, smoothly draw the curve that connects them, forming a circle.
This visual representation helps in understanding the geometric properties of the circle better.
Radius
The radius of a circle is the distance from the center to any point on the circle. It's a crucial part of its equation.
In the standard form, \[ (x - h)^2 + (y - k)^2 = r^2 \] the \( r \) represents the radius.
In our example, the radius is given as 7.
This means that every point on the circle is exactly 7 units away from the center \( (-5, -2) \).
When graphing, this uniform distance helps in marking points around the center to create the circle shape.
Understanding the radius helps solve many problems related to circles, such as calculating the area or circumference.
Center
The center of a circle is one of its most defining features.
In the standard form equation \[(x - h)^2 + (y - k)^2 = r^2 \], the \( h \) and \( k \) values determine the center's coordinates \( (h, k) \).
For our circle, the center is at \( (-5, -2) \).
This point is vital for graphing the circle because it is the starting point for marking the radius.
By knowing the center, you can easily understand the circle's position in the coordinate plane.
It also helps in comparing and translating circles, finding distances, and solving various geometric problems related to circles.

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