Posted by Anonymous on Thursday, February 14, 2013 at 11:24am.
find the equation of circle
if the line 2x  y +1 = 0 touches the circle at the point (2,5) and the center of the circle lies on the line x+y 9 =0

mathsurgently needed  Steve, Thursday, February 14, 2013 at 11:36am
If the line touches the circle, then the line from (2,5) to the center is perpendicular to the line y=2x+1, so it has slope 1/2.
So, the line through (2,5) with slope 1/2 is
y5 = 1/2 (x2)
y = 1/2 x + 6
This line passes through the center of the circle, so
1/2 x + 6 = 9x
x=6
(6,3) is the center of the circle.
The radius is the distance from (6,3) to (2,5) = √20
The circle is thus
(x6)^2 + (y3)^2 = 20
Just as a final check:
The distance from (6,3) to the line 2xy+1 = 0 is 2*6  3 + 1/√5 = 10/√5 = 2√5 = √20
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