The equation of the bisector of the acute angle between the lines $3x - 4y + 7 = 0$ and $12x + 5y - 2 = 0$ is

Let $A, B, C$ be three points in the $xy$-plane,whose position vectors are given by $\sqrt{3} \hat{i} + \hat{j}$,$\hat{i} + \sqrt{3} \hat{j}$,and $a \hat{i} + (1 - a) \hat{j}$ respectively with respect to the origin $O$. If the distance of the point $C$ from the line bisecting the angle between the vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is $\frac{9}{\sqrt{2}}$,then the sum of all the possible values of $a$ 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

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:

The straight line $x+y+1=0$ bisects an angle between a pair of lines,of which one is $2x-3y+4=0$. Then the equation of the other line in that pair is

The equation of the bisector of the acute angle between the lines $3x - 4y + 7 = 0$ and $12x + 5y - 2 = 0$ is