Lagrange Multipliers Question

Question

Find the points on $$ 4x^2+ 9y^2= 36 $$ closest and farthest from $P(1,1)$. I some how ended up with a quartic equation and it has complex roots. I don't know what went wrong.

Answer 1

The function you are minimizing/maximizing here is the distance function given by $$d(x,y) = \sqrt{(x-1)^2 + (y-1)^2}$$. WLOG, one could minimizing/maximizing the function $d^2=D$ instead since then the algebra is simpler. Define $g(x,y) = 4x^2 + 9y^2 - 36$ and let $\lambda$ be the Lagrange multiplier. Using method of Lagrange multiplier, you need to solve the following system of equations together with the constraint $g(x,y)=0$. \begin{alignat*}{3} 2(x-1) & = D_x && = \lambda g_x && = 8x.\\ 2(y-1) & = D_y && = \lambda g_y && = 18y.\\ & g(x,y) && = 0. \end{alignat*}