Let `A(1, 2) and B(-2,-4)` be two fixed points. A variable point P.is chosen on the straight line `y=x` such that perimeter of `DeltaPAB` is minimum. Find coordinates of P.

Explanation

Sine distance AB is fixed so for minimizing the perimeter of `DeltaPAB` we basically have to minimize `(PA +PB)`
Let A be the mirror image of A in line y=x (se figure).
`F(P) =PA +PB`
`F(P) =PA + PB`
But for `DeltaPAB`

`PA + AB ge A'B` and equality and when P'A and B becomes collinear. Thus for minimum path length point P is that special point for which PA and PB become incident and reflected rays with respect to the mirror y=x .
Equation of line joining A nad B is y =2x intersection of this line with y=x is the point p.
Hence p= (0,0)

Note : Above concept is very useful because such problems become very lengthy by making perimeter as a function of position of P and then minimizing it.