The lines $y = -\frac{3}{2} x$ and $y = -\frac{2}{5} x$ intersect the curve $3x^2 + 4xy + 5y^2 - 4 = 0$ at the points $P$ and $Q$ respectively. The tangents drawn to the curve at $P$ and $Q$:

$A$ chord through the point $(1,-2)$ cuts the curve $3x^2-y^2-2x+4y=0$ at $P$ and $Q$. If $PQ$ subtends an angle $\theta$ at the origin,then $\theta$ equals (in $^{\circ}$)

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

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

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 $(h^2-ab)-2gf=$

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: