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

Which of the following equations does not represent a parabola?

The equation $x^2 + 4xy + y^2 + 2x + 4y + 2 = 0$ represents

The equation $16 x^2+y^2+8 x y-74 x-78 y+212=0$ represents

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-

Consider the lines $L_1$ and $L_2$ defined by $L_1: x \sqrt{2} + y - 1 = 0$ and $L_2: x \sqrt{2} - y + 1 = 0$. For a fixed constant $\lambda$,let $C$ be the locus of a point $P$ such that the product of the distance of $P$ from $L_1$ and the distance of $P$ from $L_2$ is $\lambda^2$. The line $y = 2x + 1$ meets $C$ at two points $R$ and $S$,where the distance between $R$ and $S$ is $\sqrt{270}$. Let the perpendicular bisector of $RS$ meet $C$ at two distinct points $R^{\prime}$ and $S^{\prime}$. Let $D$ be the square of the distance between $R^{\prime}$ and $S^{\prime}$.
$(1)$ The value of $\lambda^2$ is
$(2)$ The value of $D$ is