If $ax^2-xy-3y^2-5x+20y+c=0$ represents a pair of lines passing through the point $(2,3)$,then $a-c=$

If the angles made by the lines represented by the equation $ax^2 + 2hxy + by^2 = 0$ with the $X$-axis are $\alpha$ and $\beta$, then $\tan(\alpha + \beta)$ is

If $P_1$ and $P_2$ are perpendicular distances (in units) from point $(2, -1)$ to the pair of lines $2x^2 - 5xy + 2y^2 = 0$,then the value of $P_1 P_2$ is

$PQR$ is a right-angled isosceles triangle with the right angle at $P(2, 1)$. If the equation of the line $QR$ is $2x + y = 3$,then the equation representing the pair of lines $PQ$ and $PR$ is:

The value of $\lambda$ such that $\lambda x^2-10 x y+12 y^2+5 x-16 y-3=0$ represents a pair of straight lines is:

If $s$ and $p$ are respectively the sum and the product of the slopes of the lines $3x^2 - 2xy - 15y^2 = 0$,then $s:p$ is equal to