The locus of the point $P(\mathbf{r})$ which encloses a triangle $A B P$ of area 1 sq. unit with the fixed points $A(\hat{\mathbf{i}})$ and $B(\hat{\mathbf{j}})$ is

Let $A B C D$ be a parallelogram and $E$ be the mid-point of $A B$. If $P$ is the point of intersection of $D E$ and $A C$, then $\frac{D P}{P E}+\frac{A P}{P C}=$

A vector $\mathbf{a}$ has components $2 p$ and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If $\mathbf{a}$ has components $p+1$ and 1 with respect to the new system, then

Let $A(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})$ and $B(13 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+9 \hat{\mathbf{k}})$ be two points on a line $L . C$ and $D$ be the points on $L$ on either side of $A$ at distance of 9 and 6 units respectively and $C$ lies between $A$ and $B$. Then position vectors of $C$ and $D$ are respectively