Maximum and minimum distance from the origin 
Find the maximum and minimum distances from the origin to the curve $5x^3+6xy+5y^2-8=0$

My attempt:
We have to maximise and minimise the following function $x^2+y^2$ with the constraint that $5x^3+6xy+5y^2-8=0$. 
Let
$$F(x,y)=x^2+y^2+\lambda(5x^2+6xy+5y^2-8)$$
$$\frac{\delta F(x,y)}{\delta x}=2x+\lambda(10x+6y)$$
and
$$\frac{\delta F(x,y)}{\delta y}=2y+\lambda(6x+10y)$$
Multiplying the 2 equations by y,x respectively and subtracting I get
$$\lambda(y^2-x^2)=0$$
Hence $$y=x$$
Substituting $x=y$ in $5x^3+6xy+5y^2-8=0$, I get the $x=\pm \frac{1}{\sqrt2}$ and $y=\pm \frac{1}{\sqrt2}$ . Now I am stuck. Both the points corresponds to only one distance. Did I do something wrong?

You have found the points that, under the contraint $5x^2+6xy+5y^2−8=0$, minimise $x^2+y^2$and therefore minimise the euclidean distance between the origin and the point $(x,y)$ that is 
$$d_{euclidean}\left( (0,0), (x,y) \right) \doteq \sqrt{(x-0)^2+(y-0)^2} = \sqrt{x^2+y^2}$$ 
So your minimum distance is the distance of $\left(+\frac{1}{\sqrt{2}}, +\frac{1}{\sqrt{2}}\right)$ from the origin (the same with $\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$. You have found the minimum distance: $$\sqrt{x^2+y^2}=\sqrt{\left( -\frac{1}{\sqrt{2}} \right)^2+\left( -\frac{1}{\sqrt{2}} \right) ^2}=\sqrt{\frac{1}{2}+\frac{1}{2}}=\sqrt{1}= 1$$
As lsp said, from $\lambda (y^2-x^2)$ you find an other possibility: $y=-x$ from which you get the points $\left(-\sqrt{2}, +\sqrt{2}\right)$ and $\left(+\sqrt{2}, -\sqrt{2}\right)$. Those points are the furthest from the origin as you will see when computing the distance to the origin: $$\sqrt{x^2+y^2}=\sqrt{\sqrt{2}^2+\sqrt{2}^2}=\sqrt{2+2}=\sqrt{4}= 2$$
