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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.

 

Please help soon. Thanks! :)

 Mar 26, 2020
 #1
avatar+23566 
+2

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
avatar+1956 
-1

I like your way, EP. I think it's pretty clear! NICE!

CalTheGreat  Mar 26, 2020
 #3
avatar+111321 
+1

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

 

 

cool cool cool

 Mar 26, 2020
edited by CPhill  Mar 26, 2020
edited by CPhill  Mar 26, 2020
 #4
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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. smiley Stay safe and healthy!!!

 Mar 26, 2020

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