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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
3 years, 1 month 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 \( \frac{|Ax_{1}+By_{1}+C|}{\sqrt{A^2+B^2 }} \)? Lucas Chen · 3 years, 1 month ago

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In general, you can always do this, but it often requires calculus. Samir Khan · 3 years, 1 month ago

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