If $P$ is a variable point which is at a distance of 2 units. from the line $2 x-3 y+1=0$ and $\sqrt{13}$ units from the point $(5,6)$, then the equation of the locus of $P$ is

If the equation $3 x^2+4 y^2-x y+k=0$ is the transformed equation of $3 x^2+4 y^2-x y-5 x-7 y+2=0$ after shifting the origin to the point $(\alpha, \beta)$ by the translation of axes, then $\alpha+\beta-k=$

If the intercept of a straight line $L$ made between the straight lines $5 x-y-4=0$ and $3 x+4 y-4=0$ is bisected at the point $(1,5)$, then the equation of $L$ is

$A$ line $L$ passes through the point $P(1,2)$ and makes an angle of $60^{\circ}$ with $O X$ in the positive direction. $A$ and $B$ are two points lying on $L$ at a distance of 4 units from $P$. If $O$ is the origin, then the area of $\triangle O A B$ is