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The circles \(x^2 + y^2 = 4\) and \((x - 3)^2 + (y - 5)^2 = 25\) intersect in two points, say \(A\) and \(B\). Find the slope of line segment \(\overline{AB}\).

 

Thanks!

 Mar 22, 2020

Best Answer 

 #4
avatar+23566 
+2

x^2 + y^2 -4 = 0     and    (x-3)^2 + (y-5)^2 - 25 = 0   equate the two and expand and simplify to get

 

6x + 10y-13 = 0    this is a line

10 y = -6x + 13

y = -6/10 x + 13      slope = -6/10 = -3/5    (I used desmos to look this over graphically to verify this solution)

 

Just as Chris found ! 

 Mar 22, 2020
 #1
avatar+111321 
+1

x^2 + y^2   = 4

(x-3)^2 + (y -5)^2  = 25

 

Expand the second equation

 

x^2 - 6x + 9 + y^2 - 10y + 25   = 25       sub the first equation into this for x^2, y^2

 

4  - 6x + 9 - 10y  + 25  = 25       simplify

 

13 - 6x - 10y  =  0

 

10y  = 13 - 6x

 

y =   [ 13 - 6x ]  / 10   sub this  into the first equation fo y

 

x^2  + ( [ 13  - 6x ] / 10)^2    = 4        multiply through  by 10^2  and simplify

 

100x^2 + 169 - 156x + 36x^2 = 400

 

136x^2  - 156x  - 231  = 0

 

Solving this for  x  produces

 

x = ( 39 + 25√15)  / 68          and     x  =  ( 39  - 25√15)  /68

 

And  using a little  technology   [ because of the messy math  ]  when  x  holds the  first  value, y = [ 13 - 6x ] /10  =

 

(65  - 15√15) / 68

 

And  when  x holds the second value, y =   [65 + 15√15) /68

 

So  letting A =  [  ( 39  - 25√15)  /68  ,   ( 65 +15√15)  /68  ]   and  B = [  ( 39  + 25√15)  /68 ,  ( 65  - 15√15)  /68

 

We can ignore the denominators when calculating the  slope.....so we have

 

[ 65 - 15√15]  -  [ 65 +  15√15  ]               -30√15               -3

_________________________  =         ________  =     ____

[39 + 25√15 ]  - [ 39 -  25√15 ]                 50√15                 5

 

 

 

 

cool cool  cool

 Mar 22, 2020
 #2
avatar
+1

Thanks very much. Great explanation! I love this site and the things you guys do! I miss reading MAD Magazine Lol.wink

 Mar 22, 2020
 #3
avatar+111321 
0

Hey....I miss it, too    !!!!!

 

Thanks for the thanks   !!!!!!

 

 

 

cool cool cool

CPhill  Mar 22, 2020
 #4
avatar+23566 
+2
Best Answer

x^2 + y^2 -4 = 0     and    (x-3)^2 + (y-5)^2 - 25 = 0   equate the two and expand and simplify to get

 

6x + 10y-13 = 0    this is a line

10 y = -6x + 13

y = -6/10 x + 13      slope = -6/10 = -3/5    (I used desmos to look this over graphically to verify this solution)

 

Just as Chris found ! 

ElectricPavlov Mar 22, 2020
 #5
avatar
0

Thanks much!!! You guys are great.

Guest Mar 22, 2020
 #6
avatar+111321 
0

Great job, EP

 

Much more elegant than my  "mountain out of a mole hill " answer    !!!!!

 

 

cool cool cool

CPhill  Mar 23, 2020

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