Find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$ How to find the minimum distance between the curves $y^2-xy-2x^2 =0$ and $y^2=x-2$ 
Let $y^2-xy-2x^2 =0...(1)$ and $y^2=x-2...(2)$
In equation (1) coefficient of $x^2 =-2; y^2=1, 2xy =\frac{-1}{2}$
We know that a second degree equation where $ab-h^2 =0 $  represent a parabola, 
$ab-h^2>1$ represent an ellipse 
$ab-h^2 <0$ represent a hyperbola 
Here $ab-h^2 <0$ therefore equation (1) represents hyperbola and equation (2) is parabola. ( where a,b, h are coefficients of $x^2,y^2,xy$ respectively.
Now how to get the minimum distance between the two curves please suggest .. thanks. 

Let 
$$
f(x,y) = y^2-xy-2x^2\\
g(x,y) = y^2-x+2.
$$
The segment of minimum distance has the direction of the normal line to both the conics. The normal line has the direction of the gradient. Hence you could try to solve the following system:
$$
\begin{cases}
\nabla f((x,y) + \lambda \nabla g(x,y)) = \mu\nabla g(x,y)\\
f((x,y)+\lambda\nabla g(x,y)) = 0\\
g(x,y) = 0
\end{cases}
$$
where
$$
\nabla f(x,y) = (-y-4x, 2y-x)\\
\nabla g(x,y) = (-1,2y)
$$
