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The line 𝑥 + 2𝑦 = 0 meets the circle x^2 + y^2 − 4𝑥 + 4𝑦 = 0 at A and B. Find the length of AB.

 Nov 24, 2020
 #1
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x + 2y  =  0    →  x = -2y

 

Sub this into  the equation of the circle for x

 

(-2y)^2  + y^2  -4(-2y)  + 4y   =  0   simplify

 

4y^2  + y^2  + 8y + 4y   = 0

 

5y^2  + 12y  = 0      factor

 

y (5y  + 12) =  0

 

Set  the factors to 0 and solve

 

y   = 0                            5y + 12  = 0

x = -2(0)  =0                  5y  = -12

                                        y = -12/5

                                         x=  -2(-12/5)  = 24/5

 

A = (0,0)    B  =( 24/5, -12/5)

 

AB =  sqrt [  (24/5)^2  + (-12/5)^2  ]  =    (1/5)  sqrt  [ 576 + 144 ] =  (1/5) sqrt (720)  =  (12/5)*sqrt (5)  units

 

See the graph here : https://www.desmos.com/calculator/chrvtjmvty

 

 

cool cool cool

 Nov 24, 2020

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