# 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
4 years, 7 months ago

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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 $$\frac{|Ax_{1}+By_{1}+C|}{\sqrt{A^2+B^2 }}$$?

- 4 years, 7 months ago

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

- 4 years, 7 months ago