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,x232) (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 d2=(x232)2 d^2=(\frac{x^2-3}{\sqrt{2}})^2 . Simplifying this further, we substitute x2 x^2 with a a and get a+(a32)2=a+a223a+92=a222a+92 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(ab)2+c a(a-b)^2+c , where c c is the minimum value. So d2=(a2)2+52 d^2=(a-2)^2+\frac{5}{2} . The minimum of d2=52 d^2=\frac{5}{2} . Therefore, the minimum of d=102 d=\boxed{\frac{\sqrt{10}}{2}} .

Note by Lucas Chen
7 years, 4 months ago

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Isn't it interesting how the shortest distance from a point (x1,y1) (x_{1},y_{1}) and line y=mx+b y=mx+b is y1mx1bm2+1 \frac{|y_{1}-mx_{1}-b|}{\sqrt{m^2+1}} ?

Or if you have the points (x1,y1) (x_{1},y_{1}) , and the line Ax1+By1+C=0 Ax_{1}+By_{1}+C=0 , the shortest distance between them is Ax1+By1+CA2+B2 \frac{|Ax_{1}+By_{1}+C|}{\sqrt{A^2+B^2 }} ?

Lucas Chen - 7 years, 4 months ago

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

Samir Khan - 7 years, 4 months ago

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