closest-distance-point-6-9-to-curve-y-x-2-12x-32-mister-w-your-method-can-be-applied-to-this-problem-

Question Number 83520 by john santu last updated on 03/Mar/20

closest distance point (6, 9)

to curve y = x^{2} - 12 x + 32 .

mister w your method can be

applied to this problem ?

Answered by john santu last updated on 03/Mar/20

\Rightarrow y = {(x - 6)}^{2} - 4

{(x - 6)}^{2} = y + 4

the distance d = \sqrt{{(x - 6)}^{2} + {(y - 9)}^{2}}

d^{2} = y + 4 + y^{2} - 18 y + 81

y^{2} - 17 y + 85 - d^{2} = 0

tangency Δ = 0

17^{2} - 4 (85 - d^{2}) = 0

d = \sqrt{\frac{340 - 289}{4}} = \frac{\sqrt{51}}{2}

Commented by jagoll last updated on 03/Mar/20

compare with differential

d = \sqrt{{(x - 6)}^{2} + {(y - 9)}^{2}}

= \sqrt{y + 4 + {(y - 9)}^{2}}

let f (y) = y + 4 + {(y - 9)}^{2}

f^{'} (y) = 1 + 2 (y - 9) = 0

y - 9 = - \frac{1}{2} \Rightarrow y + 4 = 13 - \frac{1}{2}

y + 4 = \frac{25}{2} \Rightarrow d_{\min} = \sqrt{\frac{25}{2} + \frac{1}{4}}

d_{\min} = \frac{\sqrt{51}}{2}

Commented by mr W last updated on 03/Mar/20

g o o d!

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