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Chapter 13: Problem 30

Write the center-radius form of each circle described. Then graph the circle. $$ x^{2}+y^{2}-8 x-12 y+3=0 $$

Short Answer

Expert verified
(x - 4)^2 + (y - 6)^2 = 49, center: (4, 6), radius: 7

Step by step solution

01

Group and reformat

First, group the x and y terms separately and move the constant term to the right side of the equation.display $$ x^2 - 8x + y^2 - 12y = -3 $$
02

Complete the square for x

To complete the square for the x terms, take half of the coefficient of x, show within the text $$ -8, \text{ then square it: } (\frac{-8}{2})^2 = 16. $$ Add and subtract 16 to the x terms. The equation now becomes: $$ x^2 - 8x + 16 + y^2 - 12y = -3 + 16 $$
03

Complete the square for y

Repeat the process for the y terms. Take half of the coefficient of y, show within then text $$ -12, \text{ square it: } (\frac{-12}{2})^2 = 36. $$ Add and subtract 36 to the y terms. The equation now is: $$ x^2 - 8x + 16 + y^2 - 12y + 36 = -3 + 16 + 36 $$
04

Simplify and write in center-radius form

Simplify the equation and rewrite each perfect square trinomial as a binomial square. The equation becomes: $$ (x - 4)^2 + (y - 6)^2 = 49 $$ , which is in the center-radius form of a circle. The center is show $$ (4, 6) $$ and the radius is radical $$ \frac{\root{2}{49}} $$ which is 7.
05

Graph the circle

Sketch a coordinate plane. Plot the center of the circle at (4, 6). From the center, measure 7 units in all directions (up, down, left, right, and diagonally). Draw a smooth curve to represent the circle.

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Key Concepts

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

completing the square
To rewrite a quadratic equation in a more useful form, we often use a method called 'completing the square'. This technique transforms the general form of a quadratic equation into a perfect square trinomial, making it easier to manipulate and understand.

Consider the exercise where we started with the equation: \( x^2 + y^2 - 8x - 12y + 3 = 0\).

We grouped the x and y terms separately and moved the constant to the right side, yielding: \( x^2 - 8x + y^2 - 12y = -3\).

To complete the square for the x terms:
  • Take half of the coefficient of x, which is -8.
  • Square this number: \( (\frac{-8}{2})^2 = 16\).
  • Add and subtract 16 within the x terms: \( x^2 - 8x + 16 \).

Do the same for the y terms:
  • Take half of the coefficient of y, which is -12.
  • Square this number: \( (\frac{-12}{2})^2 = 36\).
  • Add and subtract 36 within the y terms: \( y^2 - 12y + 36 \).
By performing these steps, the equation changes to: \( x^2 - 8x + 16 + y^2 - 12y + 36 = -3 + 16 + 36\). We can now simplify it further.
equation of a circle
An important form of a circle's equation in algebra is the center-radius form. It makes understanding and graphing circles much more straightforward.

To put the equation into center-radius form, let's simplify the completed square terms from our previous section:
The equation was: \( x^2 - 8x + 16 + y^2 - 12y + 36 = 49 \).
We recognize each quadratic trinomial as a perfect square binomial:
  • \( x^2 - 8x + 16 \rightarrow (x - 4)^2 \)
  • \( y^2 - 12y + 36 \rightarrow (y - 6)^2 \)
Rewriting the equation, we have: \( (x - 4)^2 + (y - 6)^2 = 49 \).This is the center-radius form of a circle's equation.

Here's how to interpret it:
  • \( (x - h)^2 + (y - k)^2 = r^2 \), where (h, k) is the center of the circle and r is the radius.
  • In our equation, the center \((h, k)\) is \((4, 6)\) and the radius r is \( \root{2}{49}\) or 7.
graphing circles
Graphing a circle is quite simple once you have its center-radius equation. Following is the step-by-step method to graph the circle from the given equation:
First, recall the center-radius form we derived: \( (x - 4)^2 + (y - 6)^2 = 49 \).

Follow these steps:
  • Sketch a coordinate plane.
  • Plot the center of the circle at point \((4, 6)\).
  • Measure 7 units from the center in all directions (up, down, left, right, and diagonally).
  • Draw a smooth curve to form the circle.
By ensuring equal measurement from the center in every direction, you accurately represent the circle with radius 7. This visual technique helps you see how the algebraic specifications of the circle translate into a geometric form.

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Most popular questions from this chapter

Graph each system of inequalities. \(3 x-4 y \geq 12\) \(x+3 y>6\) \(y \leq 2\)

Graph each system of inequalities. \(3 x-y>-6\) \(4 x+3 y>12\)

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{r} x^{2}+6 y^{2}=9 \\ 4 x^{2}+3 y^{2}=36 \end{array} $$

Graph each inequality. \(x \leq-y^{2}+6 y-7\)

Graph each generalized square root function. Give the domain and range. $$ f(x)=-\sqrt{36-x^{2}} $$
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