Question

The coordinates of a point $P$ on the line $2x-y+5=0$ such that $|PA-PB|$ is maximum where $A$ is $(4,-2)$ and $B$ is $(2,-4)$ will be

• A. $(11,27)$
• B. $(-11,-17)$
• C. $(-11,17)$
• D. $\left(0.5\right)$

Answer 1

The correct option is B $(-11,-17)$

$⟶$ let $\overrightarrow{P}=x\hat{i}+y\hat{j},\overline{PA}=(4-x)\hat{i}+(-2-y)\hat{j}$

$\overline{PB}=(2-x)\hat{i}+(-4-y)\hat{j}$

let $z=|\overline{PA}-\overline{PB}{|}^2=(\overline{PA}-\overline{PB})=|\overline{PA}{|}^2+|\overline{PB}{|}^2-2\overline{PA}.\overline{PB}$

$=(x-4{)}^2+(y+2{)}^2+(x-2{)}^2+(y+4{)}^2-2(4-x)(2-x)+(2+y)(4+y)$

$=2x^2+2y^2-12x+18y+40-2(x^2+8-6x+y^2+8+6y)$

$=6y+24$

$|PA-PB|$ is maximum for maximum $y$,

out of $4$ options, option $\left(A\right)$ has largest $y$.