Suppose that $$A1$$,$$2$$ and $$B3$$,$$4$$ be two points and P be a point on $$y=x$$ such that $$PA+PB$$ is minimum then the point P

Question

Suppose that $$A1$$,$$2$$ and $$B3$$,$$4$$ be two points and P  be a point on  $$y=x$$ such that $$PA+PB$$ is minimum then the  point  P

Infinity Learn · Accepted Answer

Suppose that the images of $$A(1$$,$$2)$$ and $$B(3$$,$$4)$$ in the line $$x=y$$ are C and D respectively.  The point P is the point of intersection of the lines AD and $$x=y$$, it is coincide with the  point of intersection of the lines BC and $$x=y$$ The image of $$B(3$$,$$4)$$ in the line $$y=x$$ is $$D(4$$,$$3)$$ The equation of the line joining $$D(4$$,$$3)$$ an $$A(1$$,$$2)$$ is y−$$2=2$$−$$31$$−$$4(x$$−$$1)This$$ can be simplified as y−$$2=13x$$−$$13y$$−$$6=x$$−$$1x$$−$$3y+5=0Plug$$ in $$x=y$$  to get the point of intersection of the above line and the line $$x=y$$ . It implies thatx−$$3x+5=02x=5x=52$$ Therefore, $$P=52$$,$$52$$

Suppose that $A (1, 2)$ and $B (3, 4)$ be two points and $P$ be a point on $y = x$ such that $|P A + P B|$ is minimum then the point $P$

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Suppose that A1,2 and B3,4 be two points and P be a point on y=x such that PA+PB is minimum then the point P

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Suppose that $A (1, 2)$ and $B (3, 4)$ be two points and $P$ be a point on $y = x$ such that $|P A + P B|$ is minimum then the point $P$