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The line y = (3x + 7)/2  intersects the circle x^2 + y^2 = 25 at A and B. Find the length of chord AB.

 
 Jun 23, 2022
 #1
avatar+123309 
+1

A graph might work best :

 

 

 

AB ≈   9.22 units

 

 

cool cool cool

 
 Jun 23, 2022
 #2
avatar+123309 
+1

A non-graphical method

 

The center of the circle  = (0,0)

 

The equation of the line in standard form  is

 

2y =  3x + 7

 

3x - 2y + 7  =  0                 using (0,0) for (x,y).....the distance  from  the circle's center  to the  line is  given  by

 

l  3(0)  - 2(0)  + 7 l                    7

_________________  =       _________        (1)

     sqrt  [ 3^2 + 2^2]              sqrt (13)

 

 

We can  form  a right  triangle  with a hypotenuse of 5, one  leg  = (1)   and the  other leg 1/2 the  chord length

 

So......the length of the  whole  chord is

 

2sqrt   [ 5^2 - (7/sqrt(13))^2 ]  =

 

2sqrt [  25  -  49/13 ]  =

 

2sqrt [ 276/13 ] ≈   9.215 =  9.22 (rounded to the  nearest hundreth )

 

 

cool cool cool

 
 Jun 23, 2022

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