Question
Find the equations of the 2 circles
which contain the point
, have
ther centres on the line
and touch the xaxis as a tangent.
which contain the point
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:
-
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 .
- Point on the Circle: The circle must pass through the point
-
Equation of the Circle:
The general equation of a circle with centerand radius is: Since the circle is tangent to the x-axis, we have. Therefore, the equation becomes: -
Substitute the Point
:
Plugging in the pointinto the circle’s equation: Expanding and simplifying:Using, substitute : Solving the quadratic equation:Therefore, the correspondingvalues are and . -
Form the Equations of the Circles:
- First Circle:
- Second Circle:
- First Circle:
-
Verification:
Both circles:- Have centers on the line
. - Are tangent to the x-axis.
- Pass through the point
.
- Have centers on the line
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
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Extra Insights
To find the equations of the circles, we follow these steps:
-
Center on the Line: The centers of the circles lie on the line defined by
, which we can express as . -
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 . -
Circle Equation: The equation of a circle with center
and radius is given by: Since, we can rewrite this as: -
Substituting Center Coordinates: As
from our line equation, we substitute in our circle equation: -
Point on Circle: The point
lies on the circle, therefore it satisfies the circle equation: Simplifying gives: -
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. -
Finding Corresponding k: For each
, we get the corresponding . -
Writing Circle Equations: Substitute
and values into the circle equation format to write the equations.
Final equations of the circles can be expressed as:
- Circle 1 centered at
- 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!