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# minimum

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The real numbers x and y satisfy (x - 3)^2 + (y - 4)^2 = 18.  Find the minimum value of x^2 + y^2.

May 28, 2021

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The real numbers x and y satisfy $$(x - 3)^2 + (y - 4)^2 = 18$$.

Find the minimum value of $$x^2 + y^2$$.

$$\text{Let the origin O=(0,0) } \\ \text{Let the center of the circle C=(3,4) } \\ \text{Let the radius of the circle r=\sqrt{18}=3\sqrt{2} } \\ \text{Let x^2+y^2=r_{\text{min}}^2 }$$

1. Distance between origin and center of the circle:

$$\begin{array}{|rcll|} \hline \overline{OC} &=& \sqrt{(0 - 3)^2 + (0 - 4)^2} \\ \overline{OC} &=& \sqrt{9+16} \\ \overline{OC} &=& \sqrt{25} \\ \mathbf{\overline{OC}} &=& \mathbf{5} \\ \hline \end{array}$$

2. $$x^2+y^2 = r^2_{\text{min}}$$

$$\begin{array}{|rcll|} \hline r_{\text{min}} &=& \overline{OC} - r \\ r_{\text{min}} &=& 5 -3\sqrt{2} \\ \hline x^2+y^2 &=& r_{\text{min}}^2 \\ x^2+y^2 &=& \left(5 -3\sqrt{2}\right)^2 \\ \mathbf{ x^2+y^2 } &=& \mathbf{ 43 -30\sqrt{2} } \\ \text{The minimum value of }~\mathbf{ x^2+y^2 } &=& \mathbf{ 0.5735931288 } \\ \hline \end{array}$$

May 28, 2021