A circle centered at (3, 2) is tangent to the line y=(x+1)/3. Find the area of the circle.
+128408 (3,2)
The line can be written as y = (1/3)(x + 1) (1)
We need to find a perpendicular line to this one passing through (3,2)
This line will have a slope of -3.....so we have
y = -3 ( x -3) + 2
y = -3x + 11 (2)
Set (1) and (2) equal to find the x coordinate of the point on the circle where these lines intersect
(1/3)(x + 1) = -3x + 11 multiply through by 3
x + 1 = -9x + 33 rearrange as
10x = 32
x = 32/10 = 16/5
And y = -3(16/5) + 11 = -48/5 + 55/5 = 7/5
So....the intersection is (16/5, 7/5)
And the distance between this point and (3,2) will be the radius of the circle...so....we can find r^2 as
(3 - 16/5)^2 + (2 - 7/5)^2 =
(-1/5)^2 + (3/5)^2 =
1/25 + 9/25 =
10/25 = 2/5 = r^2
So....the area of the circle = pi * r^2 = (2/5) pi units^2
Here's a graph : https://www.desmos.com/calculator/tklpk6vonx
