Question

Let $P$ be $(5,3)$ and a point $R$ on $y=x$ and $Q$ on the x-axis be such that $PQ+QR+RP$ is minimum. Then the coordinates of $Q$ are

• A. $(17/4,0)$
• B. $(17,0)$
• C. $(17/2,0)$
• D. $Noneofthese$

Answer 1

The correct option is A $(17/4,0)$

REF.Image.

Given

$P=(5,3)$

on x -axis

$Q=(x_1,0)$

R on $y=x$

$R=(x_2,x_2)$

Now,

$PQ+QR+RP$ is minimum

$PQ=(5-x_1,3-0);QR=(x_1-x_2,0-x_2)$

$RP=(x_2-5,x_2-3)$

& $(x_1,x_2)=|PQ|+|QR|+|RP|$

$=\sqrt{(5-x_1{)}^2+3^2}+\sqrt{(x_1-x_2{)}^2+(-x_2{)}^2}+\sqrt{(x_2-5{)}^2+(x_2-3{)}^2}$

min of $f(x_1,x_2)$

$x_2=\frac{17}{5},x_1=\frac{17}{4}$

$\therefore Q=(\frac{17}{4},0)$