$\overline{AB}$ is given. If $P$ is a point on $\overline{AB}$ such that $P \neq A$ and $P \neq B$,then $P$ divides $\overline{AB}$ from $A$ in the division ratio $\ldots \ldots \ldots \ldots$

For points $A(1, 7)$,$B(2, 4)$,and $C(k, 5)$,if $\angle A = 90^\circ$,then find the value of $k$.

The line segment joining $(0,0)$ and $(6,6)$ is divided into four congruent parts. Which of the following is one of the dividing points?

$A (1, 2)$ and $B (2, 3)$ are the given points. Find the point $C \in \overleftrightarrow{AB}$ such that $C - A - B$ and $2 AC = 3 AB$.

Ayush starts walking from his house to his office. Instead of going to the office directly,he goes to a bank first,from there to his daughter's school,and then reaches the office. What is the extra distance travelled by Ayush in reaching his office (in $, km$)? (Assume that all distances covered are in straight lines).
If the house is situated at $(2,4)$,the bank at $(5,8)$,the school at $(13,14)$,and the office at $(13,26)$,where coordinates are in $km$.

In which ratio and at which point does the $Y$-axis divide the line segment joining $A (-2, 4)$ and $B (1, 1)$ from $A$?