Question
upstudy study bank question image url

The circle Chas equation \[ x^{2}+y^{2}-10 x+6 y+30=0 \] Find the \( y \)-coordinates of the points where the circle \( C \) crosses the line with equation \( x=4 \), giving your answers as simplified surds. (3 marks)

Ask by Reeves Watkins. in the United Kingdom
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The \( y \)-coordinates are \( -3 + \sqrt{3} \) and \( -3 - \sqrt{3} \).

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

The Deep Dive

To find the points where the circle \( C \) crosses the line \( x = 4 \), we will substitute \( x = 4 \) into the equation of the circle. Starting with the equation of the circle: \[ x^{2} + y^{2} - 10x + 6y + 30 = 0 \] Substituting \( x = 4 \): \[ (4)^{2} + y^{2} - 10(4) + 6y + 30 = 0 \] Calculating \( (4)^{2} = 16 \) and \( -10(4) = -40 \): \[ 16 + y^{2} - 40 + 6y + 30 = 0 \] Now, simplify: \[ y^{2} + 6y + 6 = 0 \] Next, we can solve this quadratic equation using the quadratic formula \( y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), where \( a = 1, b = 6, c = 6 \). Calculating the discriminant: \[ b^{2} - 4ac = 6^{2} - 4(1)(6) = 36 - 24 = 12 \] Now substituting into the quadratic formula: \[ y = \frac{-6 \pm \sqrt{12}}{2(1)} = \frac{-6 \pm 2\sqrt{3}}{2} \] This simplifies to: \[ y = -3 \pm \sqrt{3} \] Thus, the \( y \)-coordinates where the circle crosses the line \( x = 4 \) are: \[ y = -3 + \sqrt{3} \quad \text{and} \quad y = -3 - \sqrt{3} \]

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