If $M$ is a point on the line $y=x$ and points $P(0,1), Q(2,0)$ are such that $P M+Q M$ is minimum, the…
If $M$ is a point on the line $y=x$ and points $P(0,1), Q(2,0)$ are such that $P M+Q M$ is minimum, the coordinates of $M$ are
$(0,0)$
$\left(\frac{13}{17}, \frac{13}{17}\right)$
$\left(\frac{2}{3}, \frac{2}{3}\right)$
$\left(\frac{31}{7}, \frac{31}{7}\right)$
Solution
Given,
$
\begin{aligned}
& P=(0,1) \\
& Q=(2,0)
\end{aligned}
$
Let $\quad M=(a, a) \quad[\because M$ lies on the line $y=x]$
In order to Make $P M+Q M$ is minimum
$\Rightarrow P, M, Q$ Must be collinear
$\therefore$ Slope of $P M=$ Slope of $Q M$
$
\begin{aligned}
& \frac{a-1}{a-0}=\frac{a-0}{a-2} \\
&(a-1)(a-2)=a^2 \\
& a^2-3 a+2=a^2 \\
&-3 a=-2 \\
& a=\frac{2}{3} \\
& \therefore M \text { is }\left(\frac{2}{3}, \frac{2}{3}\right)
\end{aligned}
$