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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 AB.

Mar 26, 2020

#1
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3x+15 = y    sub that in to the circle equation

x^2   +  [(3x+15)/4]^2 = 36

x^2 + [(9x^2 +90x+225)/16 ] = 36

16x^2  + 9x^2 + 90x + 225 = 576

25x^2 + 90x - 351 = 0       Use quadratic formula to find     x  =  2.35692    and  −5.95692      (these are  9/5 +- 12 sqrt5/5 )

Now you can use these values of x to find the y values by substituting into the line equation

THEN you can use the distance formula to calculate the length    sqrt  { (x1-x2)^2  + (y1-y2)^2 }

Is there an easier way???  Probably........

Mar 26, 2020
#2
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I like your way, EP. I think it's pretty clear! NICE!

CalTheGreat  Mar 26, 2020
#3
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Thanks, EP

Here's another method....albeit....maybe not any easier .....LOL!!!!!

The  line intersects  the  y axis  at  15/4  = 3.75

And it  intersects  the  x  axis at x = -5

The distance  between these two points is   sqrt  ( 5^2 + 3.75^2 ) = 6.25

Call the  distance  between  the  y intercept  and the point where the line  intersects  the  upper-most part of the  circle, a

Call the distance  from the  x intercept of the  line  and the point where the line intersects the  "lower" part of the  circle , b

The  distance  between  the  y intercept and the top of the  circle =  6 -3.75 = 2.25

And the distance  from the y intercept to the  bottom part of the  circle  =  3.75 + 6 = 9.75

Likewise......the  distance  from  the  x intercept to the  leftmost point of the  circle  =1

And  the distance  from the  x intercept to  the  rightmost part of the  circle   =11

By  the intercepting  chord  theorem  we have this system

(a) (6.25 + b)  =  (2.25) (9.75)

(a+ 6.25) (b) =  (1) (11)            simplify

6.25a  + ab  =  21.9375     (1)

6.25b  + ab  =  11        (2)          subtract these

6.25 ( a - b)  =  10.9375

(a - b)  =  10.9375/6.25

a - b =  1.75

a = b + 1.75

Subbing  this  into   (2)   we get that

(b + 1.75 + 6.25) ( b)  =11

( b + 8) b  =11

b^2  + 8b - 11 = 0

b^2  + 8b  = 11

b^2 + 8b + 16 = 11 + 16

(b + 4)^2  =  27        take the positive  root

b + 4 =  √27 = 3√3

b = 3√3  - 4

And

a = b + 1.75   =   3√3  - 2.25

So.....the  length of  chord  AB  =

b  + 6.25  + a  =

3√3 - 4 + 6.25  + 3√3  - 2.25   =

6√3  + 6.25 - 6.25  =

6√3   Mar 26, 2020
edited by CPhill  Mar 26, 2020
edited by CPhill  Mar 26, 2020
#4
0

Thank you Electric Pavlov and CPhill!!! Both your ways were good, I was kinda lost on your way CPhill. I'm not too good at piecing together something that large. I was trying something along the lines of Electric Pavlov's solution, but coudn't figure out how to finish it. Thanks both of you. Stay safe and healthy!!!

Mar 26, 2020