If x and y are numbers such that (x+9)(y-9)=0, what is the smallest possible value of x^2 + y^2
I understand its a basic question, but I fail to understand the concept. I simply put x+9=0, so x=-9. in the same way, y-9=0, so y=9
so x^2 + y^2 = 81+81=162
what am I doing wrong?
This question could me solved in this way.
(x+9)(x-9) = 0
=> xy - 81 = 9(x-y)
=> (x-y) = (xy/9) - 9
=> Squaring on both sides and cancelling equal terms
=> x^2 + y^2 = (xy)^2/81 + 81
Notice that 81 is added to (xy)^2/81, so the absolute minimum value of the above expression will be equal to 81 when x=9 and y=9 => x^2 + y^2 = 162.