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Decide whether or not the equation below has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. 9x2 + 9y? - 12x+ 12y – 17 =0 Choose the correct statement about the given equation and, if necessary, fill in the answer box(es) within your choice. O A. The graph of the equation is a circle with center (Type an ordered pair. Use integers or fractions for any numbers in the expression.) The radius of the circle is (Type an integer or a simplified fraction.) O B. The graph of the equation is the point (Type an ordered pair. Use integers or fractions for any numbers in the expression.) O C. The graph of the equation is a line. O D. The graph is nonexistent.

Decide whether or not the equation below has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. 9x2 + 9y? - 12x+ 12y – 17 =0 Choose the correct statement about the given equation and, if necessary, fill in the answer box(es) within your choice. O A. The graph of the equation is a circle with center (Type an ordered pair. Use integers or fractions for any numbers in the expression.) The radius of the circle is (Type an integer or a simplified fraction.) O B. The graph of the equation is the point (Type an ordered pair. Use integers or fractions for any numbers in the expression.) O C. The graph of the equation is a line. O D. The graph is nonexistent.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Educational Content - Understanding the Equation of a Circle** ### Determine the Nature of the Equation **Objective:** Learn how to decide whether the given equation represents a circle, and if it does, identify its center and radius. If not, describe the nature of the graph. **Given Equation:** \[ 9x^2 + 9y^2 - 12x + 12y - 17 = 0 \] **Task:** Choose the correct statement about the given equation and, if necessary, fill in the answer boxes within your choice. ### Options: **A. The graph of the equation is a circle with center \( (\text{__} , \text{__}) \).** (Type an ordered pair. Use integers or fractions for any numbers in the expression.) The radius of the circle is \( \text{__} \).\ (Type an integer or a simplified fraction.) **B. The graph of the equation is the point \( (\text{__} , \text{__}) \).**\ (Type an ordered pair. Use integers or fractions for any numbers in the expression.) **C. The graph of the equation is a line.** **D. The graph is nonexistent.** **Steps to Determine the Nature of the Equation:** 1. **Recognize the Form:** - Identify whether the equation can be written in the standard form of a circle, which is: \[ (x - h)^2 + (y - k)^2 = r^2 \] - Or check if it simplifies to another conic section or no graph at all. 2. **Complete the Square:** - To put the equation into a recognizable form, rearrange and complete the square for both \( x \) and \( y \). 3. **Analyze the Completed Form:** - Determine if the resulting form matches the equation of a circle. If it doesn't, conclude accordingly (point, line, or nonexistent). By analyzing and solving these steps, you will understand how to interpret the given quadratic equation and correctly identify its graph in the context of circle equations.
Transcribed Image Text:**Educational Content - Understanding the Equation of a Circle** ### Determine the Nature of the Equation **Objective:** Learn how to decide whether the given equation represents a circle, and if it does, identify its center and radius. If not, describe the nature of the graph. **Given Equation:** \[ 9x^2 + 9y^2 - 12x + 12y - 17 = 0 \] **Task:** Choose the correct statement about the given equation and, if necessary, fill in the answer boxes within your choice. ### Options: **A. The graph of the equation is a circle with center \( (\text{__} , \text{__}) \).** (Type an ordered pair. Use integers or fractions for any numbers in the expression.) The radius of the circle is \( \text{__} \).\ (Type an integer or a simplified fraction.) **B. The graph of the equation is the point \( (\text{__} , \text{__}) \).**\ (Type an ordered pair. Use integers or fractions for any numbers in the expression.) **C. The graph of the equation is a line.** **D. The graph is nonexistent.** **Steps to Determine the Nature of the Equation:** 1. **Recognize the Form:** - Identify whether the equation can be written in the standard form of a circle, which is: \[ (x - h)^2 + (y - k)^2 = r^2 \] - Or check if it simplifies to another conic section or no graph at all. 2. **Complete the Square:** - To put the equation into a recognizable form, rearrange and complete the square for both \( x \) and \( y \). 3. **Analyze the Completed Form:** - Determine if the resulting form matches the equation of a circle. If it doesn't, conclude accordingly (point, line, or nonexistent). By analyzing and solving these steps, you will understand how to interpret the given quadratic equation and correctly identify its graph in the context of circle equations.
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