What is the smallest distance between the origin and a point on the graph of y=1/sqrt(2) + (x^2-3)?
+14903 What is the smallest distance between the origin and a point on the graph of y=1/sqrt(2) + (x^2-3)?
Hello Guest!
\(y=\frac{1}{\sqrt{2}}+x^2-3\\ y'=2x\\ y'_{ ↲}=-\frac{1}{2}x\\ [the\ perpendicular\ on\ y '\ to\ the\ origin\ of\ the\ coordinates ] \)
\( \color{blue}x^2+\frac{1}{\sqrt{2}}-3=-\frac{1}{2}x\\x^2+ \frac{1}{2}x+\frac{1}{\sqrt{2}}-3=0\)
p q
\(x=-\frac{p}{2}\pm \sqrt{(\frac{p}{2})^2-q}\\ x=-\frac{1}{4}\pm \sqrt{(\frac{1}{4})^2-\frac{1}{\sqrt{2}}+3)}\)
\(\color{blue}x_1=1.2847\\ x_2=-1.7847\ [not\ applicable ]\)
\(y_1=\frac{1}{\sqrt{2}}+x^2-3=\frac{1}{\sqrt{2}}+1.2847^2-3\)
\(y_1=-0.64236\)
The smallest distance
\(=\sqrt{x_1^2+y_1^2}=\sqrt{1.2847^2+(-0.64236)^2}\)
\(=1.43637\)
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