The equations of the angle bisectors between the $x$-axis and $y$-axis are:

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:

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 :

Find the equation of the locus of points equidistant from the lines $3x + 4y - 11 = 0$ and $12x + 5y + 2 = 0$.

The locus of the points which are at an equal distance from $3x + 4y - 11 = 0$ and $12x + 5y + 2 = 0$ and which is near the origin is:

If the vertices of a triangle are $A(-1, -7)$,$B(5, 1)$,and $C(1, 4)$,then the equation of the angle bisector of $\angle ABC$ is: