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# The line intersects a circle centered at the origin at A and B. We know the length of chord ​ is 20. Find the area of the circle.

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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.

May 8, 2019

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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 :

May 8, 2019
edited by CPhill  May 9, 2019
edited by CPhill  May 9, 2019
edited by CPhill  May 9, 2019