The line 𝑥 + 2𝑦 = 0 meets the circle x^2 + y^2 − 4𝑥 + 4𝑦 = 0 at A and B. Find the length of AB.
+128474 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
