I'm having a hard time solving for $\lambda$ for the two equations below. $$\frac{x}{\sqrt{x^2+y^2}}=26\lambda x-10\lambda y$$ and $$\frac{y}{\sqrt{x^2+y^2}}=26\lambda y-10\lambda x.$$
I keep getting $\lambda=0$ after trying to solve the equations simultaneously. Maybe I'm doing something wrong or maybe $\lambda =0$ is the solution.
Need a succinct guideline to this.
Original question
Find the maximum and minimum value of ${\sqrt{x^2+y^2}}$ when $13x^2-10xy+13y^2=72$
Optimizing $\sqrt{x^2+y^2}$ is the same as optimizing $f(x,y)=x^2+y^2$ subject to $g(x,y)=13x^2-10xy+13y^2-72=0$. Applying Lagrange multipliers we can set up the following equations: $$\begin{cases} \frac{\partial f}{\partial x}=2x=\lambda(26x-10y)=\lambda \cdot \frac{\partial g}{\partial x}\\ \frac{\partial f}{\partial y}=2y=\lambda (26y-10x)=\lambda \cdot \frac{\partial g}{\partial y}\\ 13x^2-10xy+13y^2=72 \end{cases}$$ Solving these equations, we get $\lambda =\frac18$ or $\lambda =\frac{1}{18}$ (these equations should be easily solvable compared to your original equations). We can directly see that the maximum and minimum values of $f(x,y)$ subject to $g(x,y)=0$ are $9$ and $4$ respectively, meaning that the maximum and minimum values of $\sqrt{x^2+y^2}=\sqrt{f(x,y)}$ are $3$ and $2$.