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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 AB is 20. Find the area of the circle.

 May 8, 2021
 #1
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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

 

 

 

cool cool cool

 May 8, 2021

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