Question
upstudy study bank question image url

Question Find the center and radius of the circle represented by the equation below. \[ x^{2}+y^{2}-10 x-14 y+38=0 \] Answer Attempt i out of 2 Center: \( ( \) Radius: Submit Answer Vid

Ask by Jimenez Paul. in the United States
Mar 11,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The center of the circle is at \((5, 7)\) and the radius is \(6\).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To find the center and radius of the circle from the equation \( x^{2}+y^{2}-10 x-14 y+38=0 \), we first rewrite it in standard form by completing the square. 1. Rearranging gives us: \[ x^{2} - 10 x + y^{2} - 14 y = -38 \] 2. Completing the square for \( x \): \[ x^{2} - 10 x = (x - 5)^{2} - 25 \] 3. Completing the square for \( y \): \[ y^{2} - 14 y = (y - 7)^{2} - 49 \] 4. Substituting back, we have: \[ (x - 5)^{2} - 25 + (y - 7)^{2} - 49 = -38 \] \[ (x - 5)^{2} + (y - 7)^{2} = 36 \] Thus, the center is \( (5, 7) \) and the radius is \( \sqrt{36} = 6 \). Center: \( (5, 7) \) Radius: \( 6 \) Now, let’s dive into some history and practical applications of circles! In ancient Greece, mathematicians like Euclid and Archimedes studied circles extensively, laying down principles that govern not just geometry but also astronomy and the study of orbits. Their explorations helped pave the way for our modern understanding of circular shapes, paving the way for later innovations in engineering and architecture. Circles are not just mathematical concepts; they have real-world applications in fields like engineering and technology! For instance, when designing wheels, gears, or even in the construction of bridges, understanding the properties of circles can lead to efficient and robust designs. Whether it's ensuring that a roller coaster has the correct curve or that a roundabout facilitates smooth traffic flow, circles are crucial in countless everyday phenomena!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy