Let the line $L_1$ passing through the point of intersection of the lines $2 x+3 y-5=0$ and $4 x-5 y+7=0$ divide the line segment joining the points $(2,3)$ and $(1,-1)$ in the ratio $2: 1$. If the equation of $L_1$ is $a x+b y=1$, then $33(a-b)=$

Let $A B C$ be a triangle and $A=(1,2)$. If $x-3 y-5=0$ the and $x+5 y-9=0$ are the perpendicular bisectors of the sides $A B$ and $B C$ respectively, then the length of the side $A C$ is

Let $A(4,3,5), B(1,-2,1), C(3,2,1)$ be the vertices of a $\triangle A B C$. If the internal bisector of $\angle B A C$ meet the side $B C$ at $D$, then $C D=$

A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the coordinate axes are rotated through an angle $\alpha$ and keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$ on the new axes. Then,