SOLUTION: Find the equation of the SMALLEST circle tangent to the line {{{ x+sqrt( 3 )y-2sqrt( 3 ) = 0 }}} and the coordinate axes. Please help me solve this problem. Best regards!

Algebra ->  Circles -> SOLUTION: Find the equation of the SMALLEST circle tangent to the line {{{ x+sqrt( 3 )y-2sqrt( 3 ) = 0 }}} and the coordinate axes. Please help me solve this problem. Best regards!       Log On


   

Question 942828: Find the equation of the SMALLEST circle tangent to the line
+x%2Bsqrt%28+3+%29y-2sqrt%28+3+%29+=+0+
and the coordinate axes.
Please help me solve this problem. Best regards!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Smallest? There's only one solution.
Use the x and y intercepts.
They form a right triangle with a circle inscribed in that right triangle.
x-intercept : y=0
x-2sqrt%283%29=0
x=2sqrt%283%29
y-intercept : x=0
sqrt%283%29y-2sqrt%283%29=0
y=2
So you have a right triangle triangle with one leg, 2sqrt%283%29, and another leg, 2.
Find the hypotenuse using the Pythagorean theorem,
%282sqrt%283%29%29%5E2%2B2%5E2=H%5E2
12%2B4=H%5E2
H%5E2=16
H=4
For an inscribed circle in a right triangle,
R=%28A%2BB-H%29%2F2
R=%282sqrt%283%29%2B2-4%29%2F2
R=sqrt%283%29%2B1-2
R=sqrt%283%29-1
So the equation of the circle is,
%28x-%28sqrt%283%29-1%29%29%5E2%2B%28y-%28sqrt%283%29-1%29%29%5E2=%28sqrt%283%29-1%29%5E2
%28x-sqrt%283%29%2B1%29%5E2%2B%28y-sqrt%283%29%2B1%29%5E2=4-2sqrt%283%29