A circle centered at the origin is tangent to the line x-2y+25=0. What is the area of the circle? Find the coordinates of the point of tangency.
+128475 Distance from origin to the line =
l 0 - 2(0) + 25 l / sqrt (1^2 + 2^2) =
25 / sqrt 5 =
5sqrt (5)
This is the radius of the circle
Area = pi ( 5sqrt (5) )^2 = 125 pi units ^2
The given line will have a slope of 1/2
The equation of a line perpendicular to this passing through the origin has the equation
y = -2x (1)
The original line can be written as
y = (1/2)x + 25/2 (2)
Set (1) = (2)
-2x = (1/2)x + 25/2
-4x = x + 25
-5x = 25
x = - 5 y = -2(-5) = 10
Point of tangency = (-5 , 10)
