+0

# Graphing question

0
67
1

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

Feb 6, 2022

### 1+0 Answers

#1
+13575
0

What is the smallest distance between the origin and a point on the graph?

Hello Guest!

$$\color{blue}f(x)= y=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ y= \dfrac{x^2}{\sqrt{2}}-9\cdot \sqrt{2}\\ \color{blue}\dfrac{dy}{dx}=y\ '=\sqrt{2}\cdot x$$

$$g(x)=-\frac{1}{y\ '}(x-0)+0\\ g(x)=-\frac{1}{\sqrt{2}\cdot x}(x-0)+0\ |\ point\ direction\ equation\\ \color{blue}g(x)= -\frac{1}{\sqrt{2}}$$

$$\color{blue}g(x)=f(x)\\-\dfrac{1}{\sqrt{2}}=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ x^2-18+1=0\\ x= \sqrt{17}$$

$$x=4.1231$$

$$y=\dfrac{1}{\sqrt{2}} (x^2 - 18)\\ y=\dfrac{1}{\sqrt{2}} (17 - 18)=-\dfrac{1}{\sqrt{2}}\\$$

$$y=0.7071$$

$$d=\sqrt{x^2+y^2}=\sqrt{17+\frac{1}{2}}$$

$$d=4.1833$$

The smallest distance between the origin and a point on the graph is 4.1833 .

!

Feb 6, 2022
edited by asinus  Feb 6, 2022