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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

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