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Consider the two functions functions : $e^x$ and $\ln x$.

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I know that the shortest distance is along the common normal. But my teacher said that "both the curves are inverse of each other and symmetrical about the line $y=x$. Hence , the common normal must be perpendicular to the line $y=x$."

My problem is only regarding the last line. My teacher said it in a way that it seems obvious. But I cannot prove that why the normal must be perpendicular to the line $y=x$ also. I'm especially looking for a geometric proof.

Ted Shifrin
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An_Elephant
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1 Answers1

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Symmetry about the line $y=x$ means that reflection across that line carries one graph to the other. Since reflection preserves distance, the reflection of the line joining the two closest points must likewise be a line joining two (other?) closest points. Since it's "geometrically clear" that there can be only one such pair of points, the reflection of the line must be itself. That means that the line is perpendicular to the "mirror" $y=x$. (For example, reflection across the line sends a line with slope $m$ to a line with slope $1/m$. So, if $m=1/m$, we must have $m=1$ or $m=-1$. But our line joins two points on opposite sides of $y=x$ and thus cannot be parallel to $y=x$.)

Ted Shifrin
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  • Thanks sir. My main problem was to rigourously prove that perpendicular to mirror due to distance preservation argument. – An_Elephant Nov 30 '23 at 19:01