I would like to show the following
For $(X, Y) \sim Normal \big( (\mu_1, \mu_2), \Sigma \big) $, $$ \mathrm{Var} \big( c_1 X + c_2 Y + c_3 XY + c_4 X^2 + c_5 Y^2 \big) > 0 $$ for any, not all zero, constant $c_1, \cdots, c_5 \in \mathbb{R}$
My attempt
the statement seems equivalent to the following,
$$ c_1 x + c_2y + c_3(xy) + c_4 x^2 + c_5 y^2 = a \Rightarrow \exists! \ \ c_1 = \cdots = c_5 =0 $$
for some constant $a$ and "variables" $x, y \in \mathbb{R}$.
It seems too simple, because, by defining $(RHS) = f(x,y)$
$$ \begin{aligned} \frac{\partial}{\partial x} f(x,y) &= c_1 + c_3 y + 2c_4 x = 0 \\ \frac{\partial^2}{\partial x^2} f(x,y) &= 2c_4 = 0 \\ \frac{\partial^2}{\partial y \partial x} f(x,y) &= c_3 = 0 \\ \frac{\partial}{\partial y} f(x,y) &= c_2 + c_3x + 2c_5y = 0 \\ \frac{\partial^2}{\partial y^2} f(x,y) &= 2c_5 = 0 \\ \end{aligned} $$
so I got $c_1 = \cdots = c_5 = 0$.
But this seems quite strange, because it does not exploit any properties from the normal distribution except the fact that $x, y \in \mathbb{R}$, which is the support of normal.
Am I being right?