I am just studying mathematics and I have a task to solve it...
There are a quadratic curve: $F(x,y) =$ $a_{o}$ + $a_{1}$x + $a_{2}$y + $a_{3}x^2$ + $a_{4}y^2$ + $a_{5}xy$
and a cirle $x^2 + y^2 = 1$
All coefficients are known.
I tried to find minimum by Lagrange Multiplier:
$\nabla f(x,y)= <a_{1} + 2a_{3}x + a_{5}y, a_{2} + 2a_{4}y + a_{5}x>$
$\lambda\nabla g(x,y) = \lambda<2x,2y>$
Now i have a system of equations:
$a_{1} + 2a_{3}x + a_{5}y - 2\lambda x = 0$
$a_{2} + 2a_{4}y + a_{5}x - 2\lambda y = 0$
$x^2 + y^2 - 1 = 0$
From eq.2:
$y = \frac{{2\lambda x - 2a_{3}x - a_{1}}}{a_{5}}$
Substitute to eq. 1:
$a_{2}a_{5} + 4a_{4}\lambda x - 4a_{3}a_{2}x - 2a_{4}a_{1} - 4\lambda^2x + 4\lambda a_{3}x + 2\lambda a_{1} = 0$
And after this substituting I don't know which next step...
O.K. I tried next approach:
From $x^2 + y^2 = 1$ lets $x = cos(\theta)$ and $y = sin(\theta)$ then
$F(\theta) =$ $a_{o}$ + $a_{1}cos(\theta)$ + $a_{2}sin(\theta)$ + $a_{3}cos^2(\theta$) + $a_{4}sin^2(\theta)$ + $a_{5}cos(\theta) sin(\theta)$
$F_{(\theta)} = a_{5}(cos^2(\theta) - sin^2(\theta)) + 2(a_{4} - a_{3}) sin(\theta)cos(\theta) + a_{2}cos(\theta) - a_{1}sin(\theta) = 0$
And in this approach I can't solve equation...
Maybe you have some hints?