if minimum value of g(a;b)=(√(a^2 +b^2 −10a−10b+50))+(√(b^2 −4y+20))+ +(√(a^2 −14a+74)) is n and occurs at a=γ , b=δ, the find (n+4γ+3δ)=?


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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

	$p e r f e c t t h a n k y o u S i r$
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