Multiply both sides by r:
#r^2 = 3/9rcos(theta) - 8rsin(theta)#
Substitute #x^2 + y^2# for #r^2#:
#x^2 + y^2 = 3/9rcos(theta) - 8rsin(theta)#
Substitute x for #rcos(theta)# and y for #rsin(theta)#
#x^2 + y^2 = 1/3x - 8y#
Move the terms on the right to the left by adding #-1/3x + 8y# to both sides of the equation:
#x^2 - 1/3x+ y^2 + 8y = 0#
We are going to complete the squares so add #h^2 + k^2# to both sides of the equation:
#x^2 - 1/3x + h^2 + y^2 + 8y + k^2 = h^2 + k^2#
Set the middle term in the right side of the pattern #(x - h)^2 = x^2 -2hx + h^2# equal to the corresponding term in the equation:
#-2hx = -1/3x#
Solve for h:
#h = 1/6#
Substitute the left side of the pattern into the equation:
#(x - h)^2 + y^2 + 8y + k^2 = h^2 + k^2#
Substitute #1/6# for every h:
#(x - 1/6)^2 + y^2 + 8y + k^2 = (1/6)^2 + k^2#
Set the middle term in the right side of the pattern #(y - k)^2 = y^2 -2ky + k^2# equal to the corresponding term in the equation:
#-2ky = 8y#
Solve for k:
#k = -4#
Substitute the left side of the pattern into the equation:
#(x - 1/6)^2 + (y - k)^2 = (1/6)^2 + k^2#
Substitute -4 for every k:
#(x - 1/6)^2 + (y - -4)^2 = (1/6)^2 + (-4)^2#
Combine the right side terms:
#(x - 1/6)^2 + (y - -4)^2 = 577/36#
Write the right side as a square:
#(x - 1/6)^2 + (y - -4)^2 = (sqrt(577)/6)^2#
Done!
This a circle with a radius of #sqrt(577)/6# centered at #(1/6, -4)#
