When are the points with coordinates $(2a, 3a)$,$(3b, 2b)$,and $(c, c)$ collinear?

Given two points $A(3, 4)$ and $B(5, -2)$. If $PA = PB$ and the area of $\Delta PAB = 10$ square units,find the coordinates of point $P$.

If two vertices of an equilateral triangle are $(1, 0)$ and $(3, 0)$,and the third vertex lies in the first quadrant,find the area of the triangle.

Let the mirror image of point $A(\alpha, \beta)$ in the line mirror $x + 2y = 3$ be point $B$,and the image of $B$ in the line $3x - 2y = 5$ be $C$. If the origin is the orthocentre of triangle $ABC$ and $P(a, b)$ is a point inside the triangle such that triangles $PAB$,$PBC$,and $PCA$ have the same area,then $3(a + b)$ is:

The incentre of the triangle formed by the lines $x+y=1$,$x=1$,and $y=1$ is

Let $A(1,2)$ and $C(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides $AD$ and $BC$ are parallel to the line $7x-y=14$. If $B(\alpha, \beta)$ and $D(\gamma, \delta)$ are the other two vertices, then $|\alpha+\beta+\gamma+\delta|$ is equal to: