If $P (9a-2, -b)$ divides the line segment joining $A (3a+1, -3)$ and $B (8a, 5)$ in the ratio $3:1$,find the values of $a$ and $b$.

Find the area of a pentagon having the vertices $(4,3), (-5,6), (-7,-2), (0,-7),$ and $(3,-6)$.

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 $km$.

If $P(x, y)$ is equidistant from the points $A(a+b, b-a)$ and $B(a-b, a+b),$ then prove that $bx = ay.$

If the points are $(12, 10)$ and $(0, 8)$,then the midpoint of the line segment joining these two points is $\ldots \ldots \ldots \ldots$

If the points $A(1, 2)$,$O(0, 0)$,and $C(a, b)$ are collinear,then: