Question
upstudy study bank question image url

Find the equations of the 2 circles
which contain the point , have
ther centres on the line
and touch the xaxis as a tangent.

Ask by Santiago Mckenzie. in Ireland
Jan 29,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The equations of the two circles are:

Solution

To find the equations of the two circles that satisfy the given conditions, we’ll follow these steps:
  1. Identify the Conditions:
    • Point on the Circle: The circle must pass through the point .
    • Center on a Line: The center of the circle lies on the line , which simplifies to .
    • Tangent to the X-Axis: The circle is tangent to the x-axis, meaning the distance from the center to the x-axis is equal to the radius . This gives .
  2. Equation of the Circle:
    The general equation of a circle with center and radius is:
    Since the circle is tangent to the x-axis, we have . Therefore, the equation becomes:
  3. Substitute the Point :
    Plugging in the point into the circle’s equation:
    Expanding and simplifying:
    Using , substitute :
    Solving the quadratic equation:
    Therefore, the corresponding values are and .
  4. Form the Equations of the Circles:
    • First Circle:
    • Second Circle:
  5. Verification:
    Both circles:
    • Have centers on the line .
    • Are tangent to the x-axis.
    • Pass through the point .
Final Answer:
After expanding, the two desired circles are
(x – 6)² + (y – 1)² = 1
and
(x – 2)² + (y – 5)² = 25.

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Extra Insights

To find the equations of the circles, we follow these steps:
  1. Center on the Line: The centers of the circles lie on the line defined by , which we can express as .
  2. Touching the x-axis: For a circle to touch the x-axis, the distance from the center of the circle to the x-axis must be equal to its radius. If is the center, then the radius .
  3. Circle Equation: The equation of a circle with center and radius is given by:
    Since , we can rewrite this as:
  4. Substituting Center Coordinates: As from our line equation, we substitute in our circle equation:
  5. Point on Circle: The point lies on the circle, therefore it satisfies the circle equation:
    Simplifying gives:
  6. Expanding and Solving: After expanding both sides, we can collect like terms and solve for :
    Expanding each term results in:
    Combining and solving leads to:
    This quadratic can be simplified and solved using the quadratic formula or factoring, leading us to:
    which provides two solutions for .
  7. Finding Corresponding k: For each , we get the corresponding .
  8. Writing Circle Equations: Substitute and values into the circle equation format to write the equations.
Final equations of the circles can be expressed as:
  1. Circle 1 centered at
  2. Circle 2 centered at
In conclusion, finding those two specific circles involves some calculation but results in circles that touch the x-axis and have a common point! Happy calculating!

Latest Geometry Questions

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