Let `A P` and `B Q` be two vertical poles at points `A` and `B` respectively. If `A P=16 m ,\ \ B Q=22 m` and `A B=20 m` , then find the distance of a point `R` on `A B` from the point `A` such that `R P^2+R Q^2` is minimum.

Explanation

Let `R` be a point on AB such that
`AR=x` metres, `RB=(20-x)` metres
`therefore RP^2 = AR^2+AP^2=x^2+16^2` and `RQ^2=RB^2+BQ^2`
`=(20-x)^2+22^2`
`=x^2-40x+884`
Let `y=RP^2+RQ^2`
`therefore y=(x^2+256)+(x^2-40x+884)`
`y=2x^2-40x+1140` `frac{dy}{dx}=4x-40`
Putting `frac{dy}{dx}=0`
`implies 4x-40=0`
`implies x=10`
`frac{d^2y}{dx^2}=4`
`therefore` when x=10 `frac{d^2y}{dx^2}=4 >0 `
`therefore text(y is minimum when )x=AR=10`