To convert the equation $2x^2 + 4xy + 5y^2 - 4x - 22y + 29 = 0$ to homogeneous form,the origin is shifted to the point:

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:

Let $C$ be a curve $ax^2+2hxy+by^2+2gx+2fy+c=0$ in a Cartesian plane. By rotating the coordinate axes through an angle $\frac{\pi}{4}$ in the positive direction,if the transformed equation of $C$ is $Y^2+XY-X=0$,then $(h^2-ab)-2gf=$

What type of conic section is represented by the equation $13x^2 - 18xy + 37y^2 + 2x + 14y - 2 = 0$?

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-