Homework Help is Here – Start Your Trial Now!
Math
Geometry
Which equation represents the circle described? The radius is 2 units. The center is the same as the center of a circle whose equation is x² + y² - 8x - 6y + 24 = 0. (x + 4)² + (y + 3)² = 2 (x-4)² + (y - 3)² = 2 O (x-4)² + (y - 3)² = 2² O (x+4)² + (y + 3)² = 2²

Which equation represents the circle described? The radius is 2 units. The center is the same as the center of a circle whose equation is x² + y² - 8x - 6y + 24 = 0. (x + 4)² + (y + 3)² = 2 (x-4)² + (y - 3)² = 2 O (x-4)² + (y - 3)² = 2² O (x+4)² + (y + 3)² = 2²

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
See similar textbooks
icon
Related questions
Question
**Question: Which equation represents the circle described?** - The radius is 2 units. - The center is the same as the center of a circle whose equation is \( x^2 + y^2 - 8x - 6y + 24 = 0 \). **Options:** - \( (x + 4)^2 + (y + 3)^2 = 2 \) - \( (x - 4)^2 + (y - 3)^2 = 2 \) ⬅️ (Selected) - \( (x - 4)^2 + (y - 3)^2 = 2^2 \) - \( (x + 4)^2 + (y + 3)^2 = 2^2 \) In the above question, students are asked to determine the correct equation of a circle based on the given radius and center. The provided details help in identifying the correct equation. ### Explanation: 1. **Determine the center of the given circle equation:** The given circle's equation is \( x^2 + y^2 - 8x - 6y + 24 = 0 \). We need to rewrite this in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where (h, k) is the center and r is the radius. - Complete the square for \( x \) and \( y \): \[ x^2 - 8x \implies (x - 4)^2 - 16 \] \[ y^2 - 6y \implies (y - 3)^2 - 9 \] So, the given equation becomes: \[ (x - 4)^2 - 16 + (y - 3)^2 - 9 + 24 = 0 \] Simplify it: \[ (x - 4)^2 + (y - 3)^2 - 1 = 0 \implies (x - 4)^2 + (y - 3)^2 = 1 \] Therefore, the center of the existing circle is \((4, 3)\), and its radius is \(\sqrt{1} = 1\). 2. **Determine the new circle's equation:**
Transcribed Image Text:**Question: Which equation represents the circle described?** - The radius is 2 units. - The center is the same as the center of a circle whose equation is \( x^2 + y^2 - 8x - 6y + 24 = 0 \). **Options:** - \( (x + 4)^2 + (y + 3)^2 = 2 \) - \( (x - 4)^2 + (y - 3)^2 = 2 \) ⬅️ (Selected) - \( (x - 4)^2 + (y - 3)^2 = 2^2 \) - \( (x + 4)^2 + (y + 3)^2 = 2^2 \) In the above question, students are asked to determine the correct equation of a circle based on the given radius and center. The provided details help in identifying the correct equation. ### Explanation: 1. **Determine the center of the given circle equation:** The given circle's equation is \( x^2 + y^2 - 8x - 6y + 24 = 0 \). We need to rewrite this in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where (h, k) is the center and r is the radius. - Complete the square for \( x \) and \( y \): \[ x^2 - 8x \implies (x - 4)^2 - 16 \] \[ y^2 - 6y \implies (y - 3)^2 - 9 \] So, the given equation becomes: \[ (x - 4)^2 - 16 + (y - 3)^2 - 9 + 24 = 0 \] Simplify it: \[ (x - 4)^2 + (y - 3)^2 - 1 = 0 \implies (x - 4)^2 + (y - 3)^2 = 1 \] Therefore, the center of the existing circle is \((4, 3)\), and its radius is \(\sqrt{1} = 1\). 2. **Determine the new circle's equation:**
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer