Question

Consider the point $A\equiv (3,4)$ and $B\equiv (7,13)$.If ${}^{\prime }{P}^{\prime }$ be a point on line $y=x$ such that $PA+PB$ is minimum, then the coordinates of ${}^{\prime }{P}^{\prime }$ is

• A. $(\frac{13}{7},\frac{13}{7})$
• B. $(\frac{23}{7},\frac{23}{7})$
• C. $(\frac{31}{7},\frac{31}{7})$
• D. $(\frac{33}{7},\frac{33}{7})$

Answer 1

The correct option is C $(\frac{31}{7},\frac{31}{7})$

Let $A^{\prime }$ is the reflection of $A$ in $y=x$
Image of any coordinate will get reversed in the mirror line $y=x$
$\Rightarrow A_1=(4,3)$

now $PA+PB=A^{\prime }P+PB$ which is minimum if $A^{\prime },P,B$ are collinear. Equation of $A_{1}B$ is
$(y-3)=\frac{13-3}{7-4}(x-4)\Rightarrow 3y=10x-31$

Solving it with $y=x$ we get

$P\equiv (\frac{31}{7},\frac{31}{7})$