QUESTION: Determine maximum value of OP, O being origin of coordinates where P describes the curve $x^2 + y^2 +2z^2=5, x+2y+z=5$?
Here using lagrange method of multipliers we have two constants λ and μ , so taking partial derivative wrt x, y, and z we get three equations. When we try to eliminate either λ or μ using these three equations, since both of the constraints equal to 5, we get μ in terms of λ only.
How do we proceed from here to eliminate λ in these type of equations with two constraints both of which equal to a non-zero constant.
The answer is $\sqrt 5$.
An appropriate Lagrangian is
$$L(x,y,z,\lambda,\mu)=x^2+y^2+z^2+\lambda(x^2+y^2+2z^2-5)+\mu(x+2y+z-5).$$
The nontrivial partial derivatives are:
$$L_x = 2x+2\lambda x+\mu \\ L_y=2y+2\lambda y+2\mu \\ L_z=2z+4z\lambda+\mu.$$
If $\lambda,\mu$ were given, these conditions would allow you to find $x,y,z$: indeed $x=-\frac{\mu}{2+2\lambda},y=-\frac{2\mu}{2+2\lambda},z=-\frac{\mu}{2+4\lambda}$. One immediate consequence is that $y=2x$ (whether $\mu=0$ or not), which reduces the problem to finding the intersection of $5x^2+2z^2=5$ with $5x+z=5$. Substitution gives the quadratic equation $5x^2+2(5-5x)^2=5$. That lets you solve for $x$, obtaining two solutions. You can check whether the resulting solutions satisfy the $L_z$ condition (which we haven't used yet) and then check for optimality.