What is the relationship between covariance matrix and contour plot? 


From this video,
https://www.youtube.com/watch?v=fr5wML3y1Xk
I deduced that here, $a==b$ and $c<0$.
Here are my questions,
(1) What does it mean by "equal loss values of errors"?
(2) What happens when a-priori probabilities of $C_1$ and $C_2$ are equal or unequal? 
(3) Can you please supply me some link of study materials?
 A: Assuming that both axes are on the same scale, you have $a<b$ and $c<0$.  The contour plot for a bivariate normal distribution with the specified covariance matrix is an ellipse of the form:
$$\begin{align}
\text{const} 
&= \begin{bmatrix} x_a - \mu_a & x_b - \mu_b \end{bmatrix}
\begin{bmatrix} a & c \\ c & b \end{bmatrix}^{-1}
\begin{bmatrix} x_a - \mu_a \\ x_b - \mu_b \end{bmatrix} \\[12pt]
&= \frac{1}{ab-c^2} \begin{bmatrix} x_a - \mu_a & x_b - \mu_b \end{bmatrix}
\begin{bmatrix} b & -c \\ -c & a \end{bmatrix}
\begin{bmatrix} x_a - \mu_a \\ x_b - \mu_b \end{bmatrix} \\[12pt]
&= \frac{b (x_a - \mu_a)^2 - 2c (x_a - \mu_a) (x_b - \mu_b) + a (x_b - \mu_b)^2}{ab-c^2}, \\[6pt]
\end{align}$$
where $\mu_a$ and $\mu_b$ are the means of the two variables.  You can tell that $c<0$ because the ellipse slopes in the negative direction (i.e., from top-left to bottom-right).  You can tell that $a<b$ because the ellipse is narrower on the horizontal axis (for variable $x_a$) than on the vertical axis (for variable $x_b$).
The remaining questions pertaining to the probabilities for the classification problem is unclear to me, and more specification of the problem is needed.
