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Chapter 13: Problem 21

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ (x-5)^{2}+(y+2)^{2}=1 $$

Short Answer

Expert verified
Center: (5, -2), Radius: 1

Step by step solution

01

Identify the standard form of a circle's equation

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

Extract the center from the equation

From the given equation \((x-5)^2 + (y+2)^2 = 1\), compare it to the standard form. We can see that \(h = 5\) and \(k = -2\). Therefore, the center is \((5, -2)\).
03

Determine the radius from the equation

The given equation can be written in the form \((x-5)^2 + (y+2)^2 = 1\), where \(r^2 = 1\). Solving for \(r\), we get \(r = \sqrt{1} = 1\). Thus, the radius is 1.
04

Verify and prepare to graph the circle

With the center \((5, -2)\) and radius 1, you can plot the circle on the coordinate plane. The circle's edge will be anywhere 1 unit away from the center.

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Key Concepts

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

Center of a Circle
To find the center of a circle from its equation, it's important to recognize the formula in standard form. The equation given in the problem resembles \[(x-h)^2 + (y-k)^2 = r^2,\]where
  • \(h\) is the x-coordinate of the center of the circle,
  • \(k\) is the y-coordinate of the center of the circle.
This means the center of the circle is the point \((h, k)\). To identify these from the equation \((x-5)^2 + (y+2)^2 = 1\), compare it directly with the standard form:
The values from the equation \((x-5)^2 + (y+2)^2 = 1\) show that \(h=5\) and \(k=-2\). So, the center is at the point \((5, -2)\) on the coordinate plane.
This precise identification of the center is crucial for graphing the circle, as the entire graph is based around this central point.
Radius of a Circle
The radius of a circle is the distance from the center to any point on the circle. It's a crucial property that determines the size of the circle.
To find the radius from the standard circle equation:\((x-h)^2 + (y-k)^2 = r^2\), we look at \(r^2\), the portion of the equation after the equals sign.
For the given equation, \((x-5)^2 + (y+2)^2 = 1\), this becomes \(r^2 = 1\).
This implies that:
  • To solve for \(r\), simply calculate the square root of 1, giving \(r=1\).
So, the circle's radius is 1 unit.
Knowing the radius is just as essential as knowing the center because it tells exactly how far the circle extends in every direction from the center point.
Standard Form of a Circle Equation
Understanding the standard form of a circle equation is key to analyzing and graphing circles. This standard form is: \((x-h)^2 + (y-k)^2 = r^2\).
This formula is particularly structured to make identifying the circle's center and radius straightforward:
  • The expressions \((x-h)^2\) and \((y-k)^2\) represent the squared distances from \(x\) and \(y\) to their central coordinates, \(h\) and \(k\), respectively.
  • Solving for \(r\) gives the actual radius, by taking the square root of the equation's right-hand side.
With this form, you can easily extract the circle鈥檚 essential characteristics directly from the equation.
For example, converting a given circle equation into this form鈥攆or instance, \((x-5)^2 + (y+2)^2 = 1\)鈥攃onfirms it's already in standard form and helps us identify that the center is \((5, -2)\) with a radius of 1.
This makes it handy for not just solving problems but also graphing circles seamlessly.

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Most popular questions from this chapter

$$ \begin{aligned} &\text { Graph the system: }\\\ &\left\\{\begin{array}{l} y \leq x^{2} \\ y \geq x+2 \\ x \geq 0 \\ y \geq 0 \end{array}\right. \end{aligned} $$

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}+6 x-4 y=3 $$

If you are given a list of equations of circles and parabolas and none are in standard form, explain how you would determine which is an equation of a circle and which is an equation of a parabola. Explain also how you would distinguish the upward or downward parabolas from the left-opening or right- opening parabolas.

If you are given a list of equations of circles, parabolas, ellipses, and hyperbolas, explain how you could distinguish the different conic sections from their equations.

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. See Examples 1 through 4. $$ y=x^{2}+10 x+20 $$
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