A circle entirely in the first quadrant is tangent to the y-axis and to the line y= x/2, and the center of the circle lies on the line y = 2. Find the radius of the circle.
+128460 Since the center of the circle lies on the line y = 2 it must have the coordinates ( a, 2)
The point of tangency wth the y axis = (0,2)
The line can be written as x - 2y = 0
Using the formula for the distance from a point to a line and equating this with the distance from(a, 2) to (0,2) = a we can solve for "a" thusly
l (a) - 2(2) l
___________ = a
sqrt [ 1 + 2^2]
l a - 4 l
_______ = a
sqrt (5)
l a - 4 l = a sqrt (5)
Then either
a - 4 = asqrt (5)
a - asqrt (5) = 4
a ( 1 - sqrt (5)) = 4
a = 4 /( 1 -sqrt (5) ) reject this...."a" is negative
Or
4 - a = a sqrt (5)
4 = a + asqrt (5)
4 = a ( 1 + sqrt (5))
a = 4 / (1 + sqrt (5) ) = the radius
Here's a graph : https://www.desmos.com/calculator/gijqact2cn
