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Chapter 3: Problem 77

Find the center-radius form of the circle with the given equation. Determine the coordinates of the center, find the radius, and graph the circle. $$x^{2}+y^{2}+2 x+2 y-23=0$$

Short Answer

Expert verified
Center: (-1, -1), Radius: 5, Circle can be graphed with these values.

Step by step solution

01

Identify the Circle Equation

The given equation is \(x^2 + y^2 + 2x + 2y - 23 = 0\). This equation is in the general form of a circle equation and needs to be converted to center-radius form.
02

Rearrange and Group Terms

Rearrange the terms so similar terms are grouped together: \( (x^2 + 2x) + (y^2 + 2y) = 23 \).
03

Complete the Square for x

Take the expression for \(x\): \(x^2 + 2x\). Add and subtract 1 (the square of half the coefficient of \(x\)) to complete the square: \(x^2 + 2x + 1 - 1 = (x+1)^2 - 1\).
04

Complete the Square for y

Similarly, take the expression for \(y\): \(y^2 + 2y\). Add and subtract 1 to complete the square: \(y^2 + 2y + 1 - 1 = (y+1)^2 - 1\).
05

Substitute and Simplify

Substitute the completed squares back into the equation: \((x+1)^2 - 1 + (y+1)^2 - 1 = 23\). Simplify it to \((x+1)^2 + (y+1)^2 = 25\).
06

Identify the Center and Radius

The equation \((x+1)^2 + (y+1)^2 = 25\) is now in the center-radius form, where the center is \((-1, -1)\) and the radius is \(\sqrt{25} = 5\).
07

Graph the Circle

To graph the circle, plot the center at \((-1, -1)\), and use the radius of 5 to draw the circle around this center.

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Key Concepts

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

Center-Radius Form
The center-radius form of a circle's equation is a more intuitive way to understand and work with circles in the coordinate plane. This form is expressed as \((x-h)^2 + (y-k)^2 = r^2\), where:
  • \((h, k)\) are the coordinates of the center of the circle.
  • \(r\) is the radius of the circle.
Converting an equation from the general quadratic form, \(Ax^2 + Ay^2 + Dx + Ey + F = 0\), into this form requires completing the square. By arranging the equation into the center-radius form, one can easily identify the center and the radius, making it simpler to visualize and graph the circle on a coordinate plane.
Completing the Square
"Completing the square" is a mathematical technique used to convert a quadratic expression into a perfect square trinomial. This method is key in rearranging a circle's equation to the center-radius form. To complete the square:
  • Focus on one variable at a time, either \(x\) or \(y\).
  • The expression \(x^2 + bx\) becomes \((x + \frac{b}{2})^2 - (\frac{b}{2})^2\).
  • This process must be applied to both \(x\) and \(y\) terms.
In the context of the original exercise, you applied this method by transforming \((x^2 + 2x)\) to \((x+1)^2 - 1\) and \((y^2 + 2y)\) to \((y+1)^2 - 1\). This reshaping helps pinpoint the circle's center and radius in a visual and computationally accessible manner.
Graphing Circles
Graphing a circle with a given equation in center-radius form is a straightforward process. Here's how you can place it on the coordinate plane:
  • Identify the center \((h, k)\) and the radius \(r\) from the equation \((x-h)^2 + (y-k)^2 = r^2\).
  • Plot the center point \((h, k)\) on the plane.
  • Using the radius \(r\), mark points \(r\) units from the center in all directions 鈥攗p, down, left, and right.
  • Sketch the circle through these points, ensuring it remains equidistant from the center.
In our exercise, with a center at \((-1, -1)\) and a radius of 5, you would plot the center and then use a compass or a consistent method to mark and connect points 5 units away in all directions. This visual representation shows how centered circles manifest in a Cartesian system.

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Most popular questions from this chapter

For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverses $$-3,2$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$P=\frac{E^{2} R}{(r+R)^{2}} \text { for } R$$

Solve each equation. For equations with real solutions, support your answers graphically. $$9 x^{2}-12 x=-8$$

Solve each quadratic equation by completing the square. $$2 x^{2}+6 x-3=0$$

Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a>0, b^{2}-4 a c<0$$
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