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Question Number 140417 by mathsuji last updated on 07/May/21

i f m i n i m u m v a l u e o f

g (a; b) = \sqrt{a^{2} + b^{2} - 10 a - 10 b + 50} + \sqrt{b^{2} - 4 y + 20} +

+ \sqrt{a^{2} - 14 a + 74}

i s n a n d o c c u r s a t a = γ, b = δ, t h e f i n d

(n + 4 γ + 3 δ) = ?

Answered by mr W last updated on 07/May/21

g (a, b) = \sqrt{{(a - 5)}^{2} + {(b - 5)}^{2}} + \sqrt{{(b - 2)}^{2} + 16} + \sqrt{{(a - 7)}^{2} + 25}

= \sqrt{{(a - 5)}^{2} + {(5 - b)}^{2}} + \sqrt{4^{2} + {(b - 2)}^{2}} + \sqrt{{(7 - a)}^{2} + 5^{2}}

⩾ \sqrt{{(a - 5 + 4 + 7 - a)}^{2} + {(5 - b + b - 2 + 5)}^{2}}

= \sqrt{6^{2} + 8^{2}}

= 10

m i n i m u m n = 10 o c c u r s w h e n

\frac{a - 5}{5 - b} = \frac{4}{b - 2} = \frac{7 - a}{5} = \frac{6}{8}

\Rightarrow a = 7 - \frac{5 \times 6}{8} = \frac{13}{4} = γ

\Rightarrow b = 2 + \frac{4 \times 8}{6} = \frac{22}{3} = δ

\Rightarrow n + 4 γ + 3 δ = 10 + 13 + 22 = 45

Commented by mr W last updated on 07/May/21

m i n k o w s k i i n e q u a l i t y w a s a p p l i e d :

\sqrt{a^{2} + b^{2}} + \sqrt{c^{2} + d^{2}} + \sqrt{e^{2} + f^{2}} ⩾ \sqrt{{(a + c + e)}^{2} + {(b + d + f)}^{2}}

Commented by mathsuji last updated on 09/May/21

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