The equation $8 x^2+12 y^2-4 x+4 y-1=0$ represents

When does the equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represent an ellipse?

If curves $y = ax^2 + bx + c$ and $y = px^2 + qx + r$ do not intersect each other and $a, b, c, p, q, r \in \{1, 2, 3, 4, \dots, 10\}$,then the maximum value of $(aq - bp)^2 + (c - r)^2$ is-

To eliminate the $xy$ term from the second-degree equation $5x^2 + 8xy + 5y^2 + 3x + 2y + 5 = 0$,the axes are rotated by an angle $\theta$. What is the value of $\theta$?

In order to eliminate the first degree terms from the equation $2x^2+4xy+5y^2-4x-22y+7=0$,the point to which the origin is to be shifted is:

The number of integral values of $p$ in the domain $[-5, 5]$ such that the equation $2x^2 + 4xy - py^2 + 4x + qy + 1 = 0$ represents a pair of lines is: