use lagrange multipliers to find the minimum distance from the curve or surface to the indicated point…

Explanation:

Step1: Set up the distance formula and constraint

The distance formula from a point ((x,y)) to ((0,10)) is (d=\sqrt{(x - 0)^2+(y - 10)^2}). To simplify calculations, we can minimize (f(x,y)=x^{2}+(y - 10)^{2}) (since minimizing (d) is equivalent to minimizing (d^{2})). The constraint is (g(x,y)=(x - 2)^{2}+y^{2}-2=0). By the method of Lagrange multipliers, (\nabla f=\lambda\nabla g). We have (\nabla f=\left\langle 2x,2(y - 10)\right\rangle) and (\nabla g=\left\langle 2(x - 2),2y\right\rangle). So, the system of equations is: (\begin{cases}2x=\lambda\times2(x - 2)\2(y - 10)=\lambda\times2y\(x - 2)^{2}+y^{2}=2\end{cases}) From the first equation (2x = 2\lambda(x - 2)), we get (x=\lambda(x - 2)), so (\lambda=\frac{x}{x - 2}(x\neq2)). From the second equation (2(y - 10)=2\lambda y), we get (\lambda=\frac{y - 10}{y}(y\neq0)). Then (\frac{x}{x - 2}=\frac{y - 10}{y}), cross - multiplying gives (xy=(x - 2)(y - 10)), which expands to (xy=xy-10x-2y + 20), so (10x+2y=20) or (y = 10 - 5x).

Step2: Substitute (y = 10 - 5x) into the constraint equation

Substitute (y = 10 - 5x) into ((x - 2)^{2}+y^{2}=2): ((x - 2)^{2}+(10 - 5x)^{2}=2) Expand: (x^{2}-4x + 4+100-100x + 25x^{2}=2) Combine like terms: (26x^{2}-104x + 102 = 0) Divide by 2: (13x^{2}-52x + 51=0) Using the quadratic formula (x=\frac{52\pm\sqrt{52^{2}-4\times13\times51}}{2\times13}=\frac{52\pm\sqrt{2704 - 2652}}{26}=\frac{52\pm\sqrt{52}}{26}=\frac{52\pm2\sqrt{13}}{26}=2\pm\frac{\sqrt{13}}{13}) If (x = 2+\frac{\sqrt{13}}{13}), then (y=10-5(2+\frac{\sqrt{13}}{13})=10 - 10-\frac{5\sqrt{13}}{13}=-\frac{5\sqrt{13}}{13}) If (x = 2-\frac{\sqrt{13}}{13}), then (y=10-5(2-\frac{\sqrt{13}}{13})=10 - 10+\frac{5\sqrt{13}}{13}=\frac{5\sqrt{13}}{13})

Step3: Calculate the distance

The distance formula (d=\sqrt{x^{2}+(y - 10)^{2}}) For the point ((x = 2-\frac{\sqrt{13}}{13},y=\frac{5\sqrt{13}}{13})) (d=\sqrt{(2-\frac{\sqrt{13}}{13})^{2}+(\frac{5\sqrt{13}}{13}-10)^{2}}) (=\sqrt{4-\frac{4\sqrt{13}}{13}+\frac{13}{169}+\frac{325}{169}-\frac{100\sqrt{13}}{13}+100}) (=\sqrt{104-\frac{104\sqrt{13}}{13}+\frac{338}{169}}) Another way is to use the geometric interpretation. The center of the circle (C=(2,0)) and radius (r=\sqrt{2}). The distance between ((0,10)) and ((2,0)) is (D=\sqrt{(2 - 0)^{2}+(0 - 10)^{2}}=\sqrt{4 + 100}=\sqrt{104}) The minimum distance (d=\sqrt{104}-\sqrt{2}\approx10.20 - 1.41=8.79)

Answer:

(8.79)