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What is the smallest distance between the origin and a point on the graph of \(y = \frac{1}{\sqrt{2}} (x^2 - 18)?\)

 Feb 27, 2024
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Let (x,y) be a point on the graph of y=1/2​(x2−18). The distance between this point and the origin is given by the distance formula:

 

x2+y2​=x2+(1/2​(x2−18))2​.

 

Squaring both sides, we get:

 

x2+21​(x2−18)2=x4−4x2+81=(x2−2)2+77.

 

To minimize the distance, we want to minimize x2−2. Since the square of a real number is non-negative, its minimum value is 0. Therefore, the minimum value of x2−2 is 0, which occurs when x=±2. Substituting x=±2 back into the equation for the distance, we get:

 

sqrt(2^2 + (1/sqrt(2)((2)^2 - 18))^2) = sqrt(4 + 1/2) = sqrt(9/2) = 3/sqrt(2) = 3*sqrt(2)/2

 

Therefore, the answer is 3*sqrt(2)/2.

 Feb 29, 2024

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