The line y = (3x + 20)/5 intersects a circle centered at the origin at A and B. We know the length of chord AB is 20. Find the area of the circle.
+128475 y = (1/5)(3x + 20) → 5y = 3x + 20 → 3x - 5y + 20 = 0
We can find the distance from the center of the circle (0,0) to this line with the following
distance = l 3(0) -5(0) + 20 l 20
___________________ = ______ (1)
sqrt ( 3^2 + 5^2 ) sqrt (34)
A perpendicular line drawn from the center of the circle to the chord will bisect this chord at right angles
Call the bisection pt, P and (1) = OP
And 1/2 the chord length = 10 = PB
Using the Pythagorean Theorem, we can find the radius, r, of the circle thusly :
r = sqrt [ (PB)^2 + (OP)^2 ]
r= sqrt [ 10^2 + (20/sqrt (34) )^2 ]
r= sqrt [ 10^2 + (400)/34 ]
r ^2 = 100 + 400/34
r^2 = [3800] 34 = 1900/17
Area of circle = pi * r^2 = (1900/17) pi units^2
