The line \(y = (3x + 20)/4\) intersects a circle centered at the origin at A and B. We know the length of chord \(\overline{AB}\) is 20. Find the area of the circle.
+128474 Writing the given line in standard form, we have that
4y = 3x + 20
3x - 4y + 20 = 0
Using the equation for the distance from a point - (0,0) - to the given line, we have that
abs [ 3(0) - 4(0) + 20 ] 20
__________________ = _______ = 4 units
sqrt [ (3)^2 + (-4)^2 ] sqrt(25)
Call this distance OP........and a segment drawn from the center of a circle that is perpendicular to a chord also bisects that chord
Since OP is a perpendicular bisector of the chord....we have a right triangle with the radius of the circle as the hypotenuse (OA).....and legs of (1/2) the chord (AP) length= 10 and OP = 4
So....r^2 = (10)^2 + (4)^2 = 116
So....the area of the circle = pi * r^2 = pi * 116 = 116 pi units^2
Here's a pic :
