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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.

 May 15, 2021
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Draw the line:  y  =  (3x + 20) / 5.

 

From the origin draw a line that passes through the origin that is perpendicular to the original line.

The point of intersection of these two lines will be the center of the chord of the circle.

 

The slope of  y  =  (3x + 20) / 5  is  3/5.

The slope of the line perpendicular to this line is  -5/3.

The perpendicular line passes through the origin, so its equation is  y  =  -5/3·x.

 

The intersection of  y  =  (3x + 20) / 5  and  y  =  -5/3·x  is the point  (-30/17, 50/17).

 

The distance from this point to point A is 10.

 

The distance from this point to the origin is  sqrt(  (-30/17 - 0)2  +  (50/17)2  )  =  sqrt( 900/289 + 2500/289 )

                     =  sqrt( 3400 / 289 )

 

Now, find the distance from the origin to point A by using the Pythagorean Theorem; this will be the radius of the circle.

 May 15, 2021

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