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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.

 Nov 25, 2020
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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

 

 

cool cool cool

 Nov 26, 2020

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