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The line $$y = \frac{3x + 15}{4}$$ intersects the circle $$x^2 + y^2 = 36$$ at $$A$$ and $$B$$. Find the length of chord $$\overline{AB}$$.

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Thank you very much!

May 10, 2020

#1
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Find the points of intersection of the line  y  =  (3x + 15)/4  and the circle  x2 + y2  =  36

and then determine the distance between them.

First, using substitution to find the points:  x2 + [ (3x + 15)/4 ]2  =  36

x2 + (9x2 + 90x + 225)/16  =  36

16x2 + 9x2 + 90x + 225  =  576

25x2 + 90x - 351  =  0

Using the quadratic formula to find the values for x:

x1  =  ( -90 + sqrt( 43200 ) ) / 50

x2  =  ( -90 - sqrt( 43200 ) ) / 50

Substitue each value of x into the equation  y  =  (3x + 15)/4  to get its corresponding y-value.

Then, use the distance formula to find the distance between these two points.

May 10, 2020
#2
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Ok thank you! Those numbers look terrifying! Time to get to work!

May 10, 2020
#3
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Just asking...is there any easier way to do this problem? Computing seems like it would take a very long time.

May 10, 2020