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Chapter 2: Problem 85

Use a graphing utility to graph each circle whoseequation is given. Use a square setting for the viewing window. $$x^{2}+10 x+y^{2}-4 y-20=0$$

Short Answer

Expert verified
The circle with the given equation has center (-5, 2) and radius 7.

Step by step solution

01

Complete the square for the \(x\)-terms

Rewrite the equation in a new order: \(x^{2}+10x + y^{2}-4y = 20\). Complete the square for the \(x\)-terms, which means adding the square of half the coefficient of \(x\) (which is \((10/2)^2 = 25\)) to both sides of the equation. This gives us: \(x^{2}+10x + 25 + y^{2}-4y = 20 + 25\).
02

Complete the square for the \(y\)-terms

Now do the same for the \(y\)-terms: add the square of half the coefficient of \(y\) to both sides. In this case, that's \((-4/2)^2 = 4\). This gives us: \(x^{2}+10x + 25 + y^{2}-4y + 4 = 20 + 25 + 4\).
03

Simplify the equation

Simplify the left side of the equation by writing the completed squares as binomials squared and simplify the right side by adding the constants. This gives us: \((x + 5)^2 + (y - 2)^2 = 49\). This is a circle's equation in standard form: \((x-h)^2 + (y-k)^2 = r^2\).
04

Determine the center and radius

This gives us the center \((h, k) = (-5, 2)\) and the radius \(r = \sqrt{49} = 7\). Now these parameters can be used to graph the circle on a square grid.

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Key Concepts

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

Completing the Square
Completing the square is a technique used in algebra to rewrite quadratic equations in a way that makes them easier to solve or graph. The main idea is to form a perfect square trinomial from quadratic terms. This helps in transforming equations into a recognizable geometric form, like that of a circle.

In our exercise, when we started with the equation \( x^2 + 10x + y^2 - 4y - 20 = 0 \), the goal was to group and transform the \(x\) and \(y\) terms into squares.

**Steps for Completing the Square:**
  • Take the coefficient of the linear terms (here, \(10\) for \(x\) and \(-4\) for \(y\)).
  • Divide this coefficient by 2.
  • Square the result and add it to both sides of the equation.
Applying this, we transformed \(x^2 + 10x\) into \((x+5)^2 - 25\) and \(y^2 - 4y\) into \((y-2)^2 - 4\). This rewriting allows us to clearly see the form of a circle.
Equation of a Circle
The equation of a circle in its standard form is given by \((x - h)^2 + (y - k)^2 = r^2\). This representation is extremely helpful for easily identifying a circle's center and radius.

**Standard Form Components:**
  • \(h\) and \(k\) are the coordinates of the circle's center.
  • \(r\) is the radius of the circle.
In our transformed equation \((x + 5)^2 + (y - 2)^2 = 49\), it aligns perfectly with the standard form where:
  • \(h = -5\), \(k = 2\), making the center \((-5, 2)\).
  • \(r^2 = 49\) hence \(r = 7\).
Understanding this equation form allows us to identify key circle parameters without extensive calculation.
Graphing Utility
A graphing utility is a valuable tool for visualizing equations, particularly those of geometric figures like circles. These utilities make it easier to check your work and gain a deeper understanding of abstract mathematical concepts.

By inputting the standard form equation \((x + 5)^2 + (y - 2)^2 = 49\) into a graphing utility, you can clearly see the circle plotted with its center at \((-5, 2)\) and radius of \(7\).

**Advantages of Using a Graphing Utility:**
  • Quick and accurate graphing without manual plotting.
  • Ability to visually verify the solution.
  • Option to explore modifications and observe immediate changes.
Graphing utilities are indispensable in confirming your algebraic manipulations and ensuring the geometric interpretations align with your calculations.
Radius and Center of Circle
The radius and center are fundamental properties of a circle that define its size and position in a plane. Identifying these from the equation \((x-h)^2 + (y-k)^2 = r^2\) is straightforward once the equation is in standard form.

**Understanding the Circle鈥檚 Characteristics:**
  • The center \((h, k)\) tells you where the circle is positioned horizontally and vertically. For example, in our exercise, the center is at \((-5, 2)\).
  • The radius \(r\) is the distance from the center to any point on the circle's edge鈥攊n this case, \(r = 7\).
Knowing these parameters helps with accurate graphing and understanding the geometric nature of the circle. It allows you to visualize how the circle would appear on a coordinate plane.

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Most popular questions from this chapter

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\sqrt{x}, g(x)=x-2$$

In your own words, describe how to find the midpoint of a line segment if its endpoints are known.

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