$4x^2-5xy+4y^2=19$ with $Z=x^2+y^2$, find $\dfrac1{Z_{\max}}+\dfrac1{Z_{\min}}$ I have no idea how to approach this question at all. I've tried to find the maximum and minimum of the quadratic but i am too confused on what to do afterwards.

Without Lagrange multipliers (but with trigonometry). Use polar coordinates:
$x=\sqrt Z\cos\theta$
$y=\sqrt Z\sin\theta$
$xy=Z\sin\theta\cos\theta=\frac{Z}{2}\sin(2\theta)$
From your equation you get $\frac{1}{z}=\frac{4}{19}-\frac{5\sin(2\theta)}{38}$
As $\sin$ takes values between $-1$ and $1$:
So $\frac{1}{z_{min}}+\frac{1}{z_{max}}=2\cdot\frac{4}{19}=\frac{8}{19}$
