The equation $14x^2 - 4xy + 11y^2 - 44x - 58y + 71 = 0$ represents

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 $\theta$?

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

To remove the $xy$ term from the second-degree equation $5x^2 + 8xy + 5y^2 + 3x + 2y + 5 = 0$,the coordinate axes are rotated through an angle $\theta$. Then $\theta$ equals:

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

The centre of the conic represented by the equation $2x^2 - 72xy + 23y^2 - 4x - 28y - 48 = 0$ is