Let $P \equiv (-1, 0)$,$Q \equiv (0, 0)$,and $R = (3, 3\sqrt{3})$ be three points. The equation of the bisector of the angle $PQR$ is

Let $u \equiv ax + by + a \sqrt[3]{b} = 0$ and $v \equiv bx - ay + b \sqrt[3]{a} = 0$ where $a, b \in R$ be two straight lines. The equation of the bisectors of the angle formed by $k_1u - k_2v = 0$ and $k_1u + k_2v = 0$ for non-zero real $k_1$ and $k_2$ are:

If the line $l_1: 3y - 2x = 3$ is the angular bisector of the lines $l_2: x - y + 1 = 0$ and $l_3: \alpha x + \beta y + 17 = 0$,then $\alpha^2 + \beta^2 - \alpha - \beta$ is equal to

Let $'C'$ be the locus of the centre of circles which touch the lines $x + 2y - 5 = 0$ and $2x - 4y + 7 = 0$. If the area enclosed by the curve $'C'$ with the line $x - y - 5 = 0$ is $\frac{P^2}{2Q^2}$,where $P$ and $Q$ are relative primes,then the value of $P + Q$ is:

In a $\triangle ABC$,suppose $y=x$ is the equation of the angle bisector of $\angle B$ and the equation of the side $AC$ is $2x-y=2$. If $2AB=BC$ and the points $A$ and $B$ are $(4,6)$ and $(\alpha, \beta)$ respectively,then $\alpha+2\beta$ is equal to

The equal sides of an isosceles triangle are given by the equations $7x-y+3=0$ and $x+y-3=0$. If the slope $m$ of the third side is an integer,then $m=$