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,...
Related questions
Question
![**Problem Statement:**
Write the equation of the circle centered at \((-10, -6)\) that passes through \((-17, 16)\).
**Solution:**
The equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Given the center at \((-10, -6)\), the equation becomes:
\[
(x - (-10))^2 + (y - (-6))^2 = r^2
\]
Simplifying, we have:
\[
(x + 10)^2 + (y + 6)^2 = r^2
\]
To determine \(r\), use the distance formula between the center \((-10, -6)\) and the point \((-17, 16)\):
\[
r = \sqrt{((-17) - (-10))^2 + (16 - (-6))^2}
\]
Simplifying:
\[
r = \sqrt{(-17 + 10)^2 + (16 + 6)^2}
\]
\[
r = \sqrt{(-7)^2 + 22^2}
\]
\[
r = \sqrt{49 + 484}
\]
\[
r = \sqrt{533}
\]
Thus, the radius squared, \(r^2\), is 533. Therefore, the equation of the circle is:
\[
(x + 10)^2 + (y + 6)^2 = 533
\]
The proposed equation \( (x + 10)^2 + (y + 6)^2 = 1213 \) is incorrect. The correct equation is:
\[
(x + 10)^2 + (y + 6)^2 = 533
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d656125-4418-43a7-9497-1ff8e92c92dc%2Fd887dd0f-7a0c-4ac4-9da8-8601c7662a56%2Faycbe6h_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Write the equation of the circle centered at \((-10, -6)\) that passes through \((-17, 16)\).
**Solution:**
The equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Given the center at \((-10, -6)\), the equation becomes:
\[
(x - (-10))^2 + (y - (-6))^2 = r^2
\]
Simplifying, we have:
\[
(x + 10)^2 + (y + 6)^2 = r^2
\]
To determine \(r\), use the distance formula between the center \((-10, -6)\) and the point \((-17, 16)\):
\[
r = \sqrt{((-17) - (-10))^2 + (16 - (-6))^2}
\]
Simplifying:
\[
r = \sqrt{(-17 + 10)^2 + (16 + 6)^2}
\]
\[
r = \sqrt{(-7)^2 + 22^2}
\]
\[
r = \sqrt{49 + 484}
\]
\[
r = \sqrt{533}
\]
Thus, the radius squared, \(r^2\), is 533. Therefore, the equation of the circle is:
\[
(x + 10)^2 + (y + 6)^2 = 533
\]
The proposed equation \( (x + 10)^2 + (y + 6)^2 = 1213 \) is incorrect. The correct equation is:
\[
(x + 10)^2 + (y + 6)^2 = 533
\]
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