+48 The line y=(3x+20)/4 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 I'll admit that this one baffled me for a moment, too.....but
(1) First find the distance between the origin and the line using the formula in the other problem
The equation of the line can be re-wriiten as
4y = 3x + 20
3x - 4y + 20 = 0
l 3(0) - 4(0) + 20 l / sqrt ( 3^2 + 4^2) =
20 / sqrt 25
20 / 5 = 4
Now....if we drew a perpendicular to the line from the origin , this would meet the chord at its midpoint
And the length of the perpendicular and 1/2 the chord length form legs of a right triangle with the radius of the circle as the hypotenuse
Using the Pythagorean Theorm, we can find the radius^2 as
4^2 + 10^2 = r^2
116 = r^2
The area of the circle =
pi r^2 =
pi * 116 =
116 pi units^2
See the graph here : https://www.desmos.com/calculator/vlrvdecxml
A = ( -10.4, -2.8) B = ( 5.6, 9.2)
Checkto see that AB = 20
