# Shortest Distance

I wonder what the shortest distance from any point to any graph of an equation is.

An example problem is: Find the shortest distance between $$(0,0)$$ and $$y=\frac{x^2-3}{\sqrt{2}}$$.

Solution to Example Problem: Any random point on the graph of the equation would be $(x,\frac{x^2-3}{\sqrt{2}})$. Using the distance formula to find the distance between the origin and that graph, we simplify and get $d^2=(\frac{x^2-3}{\sqrt{2}})^2$. Simplifying this further, we substitute $x^2$ with $a$ and get $a+(\frac{a-3}{\sqrt{2}})^2= a+\frac{a^2}{2} -3a+\frac{9}{2}=\frac{a^2}{2}-2a+\frac{9}{2}$.

Now, we must complete the square, completing it in the form $a(a-b)^2+c$, where $c$ is the minimum value. So $d^2=(a-2)^2+\frac{5}{2}$. The minimum of $d^2=\frac{5}{2}$. Therefore, the minimum of $d=\boxed{\frac{\sqrt{10}}{2}}$.

Note by Lucas Chen
7 years, 4 months ago

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## Comments

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Isn't it interesting how the shortest distance from a point $(x_{1},y_{1})$ and line $y=mx+b$ is $\frac{|y_{1}-mx_{1}-b|}{\sqrt{m^2+1}}$?

Or if you have the points $(x_{1},y_{1})$, and the line $Ax_{1}+By_{1}+C=0$, the shortest distance between them is $\frac{|Ax_{1}+By_{1}+C|}{\sqrt{A^2+B^2 }}$?

- 7 years, 4 months ago

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In general, you can always do this, but it often requires calculus.

- 7 years, 4 months ago

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