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Q.
Consider the points $A (3,4)$ and $B (7,13)$. If $P$ is a point on the line $y=x$ such that $P A+P B$ is minimum, then the coordinates of $P$ are
Straight Lines
Solution:
Consider a point $A$', the image of $A$ in $y=x$.
Therefore, the coordinates of $A$' are $(4,3)$
or
(Notice that $A$ and $B$ lie on the same side with respect to $y = x$ ).
Then $PA = PA$'.
Thus, $PA + PB$ is minimum, If $PA'+ PB$ is minimum,
i.e., if $P , A ', B$ are collinear. Now, the equation of $A B$ is
$y -3=\frac{13-3}{7-4}( x -4)$
or $3 y-10 x+31=0$
It intersects $y = x$ at $(31 / 7,31 / 7)$, which is the required point $P$.
