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Answer:
1. Identify the center coordinates ((h,k):
- Since the circle is tangent to ( x = 8 ) and ( x = 14 ), the horizontal distance from the center to these lines is the radius ( r ).
- Therefore, we can write:
[tex] \[
|h - 8| = r \quad \text{and} \quad |h - 14| = r
\][/tex]
- Because the circle is tangent to both lines, the distance between the lines must be twice the radius:
[tex] \[
14 - 8 = 2r \implies r = 3
\]
[/tex]
[tex] \[
h - 8 = 3 \implies h = 11
\][/tex]
2. identify the vertical coordinate ( k ):
- Since the circle is tangent to ( y = 3 ), the vertical distance from the center to this line is also the radius ( r ):
[tex] \[
k - 3 = r \implies k - 3 = 3 \implies k = 6
\][/tex]
3. Summarize the center and radius:
[tex]\( (h, k) = (11, 6) \)[/tex]
[tex] \( r = 3 \)[/tex]
4. Write the equation of the circle:
Using the standard form
[tex]\((x-h)^2 + (y-k)^2 = r^2\), \: we \: get:[/tex]
[tex]\[
(x - 11)^2 + (y - 6)^2 = 3^2
\][/tex]
[tex] \[
(x - 11)^2 + (y - 6)^2 = 9
\][/tex]
[tex]\[
(x - 11)^2 + (y - 6)^2 = 9
\][/tex]