given-5x-12y-60-min-value-of-x-2-y-2-

Question Number 103040 by bobhans last updated on 12/Jul/20

g i v e n 5 x + 12 y = 60

m i n v a l u e o f \sqrt{x^{2} + y^{2}}

Answered by bobhans last updated on 12/Jul/20

Commented by mathmax by abdo last updated on 12/Jul/20

error sir \frac{\partial f}{\partial λ} = 5 x + 2 y - 60!

Commented by bemath last updated on 12/Jul/20

t h e e q u a t i o n 5 x + 12 y - 60 = 0 s i r

i t t y p o

Answered by ajfour last updated on 12/Jul/20

l e t x = r \cos θ, y = r \sin θ

\Rightarrow r = \sqrt{x^{2} + y^{2}}

13 (\frac{5 x}{13} + \frac{12 y}{13}) = 60

{i f \sin α = \frac{5}{13}, \cos α = \frac{12}{13}

\Rightarrow \tan α = \frac{5}{12}}

13 r (\sin α \cos θ + \cos α \sin θ) = 60

\Rightarrow 13 r \sin (α + θ) = 60

\Rightarrow r = \frac{60}{13 \sin (α + θ)}

a n d a s r > 0 \Rightarrow r_{m i n} = \frac{60}{13}

w h e n θ = 2 n π + \frac{π}{2} - \tan^{- 1} \frac{5}{12} .

Answered by maths mind last updated on 12/Jul/20

M (x, y) \in (D) : 5 x + 12 y - 60 = 0

\sqrt{x^{2} + y^{2}} = d (O, M)

m i n d (O, M) = d (O, M^{'})

M^{'} p t o j e c t i o n o f O o v e r (d)

d (0, M^{'}) = \frac{∣ 0.5 + 12.0 - 60 ∣}{\sqrt{5^{2} + 12^{2}}} = \frac{60}{13}

Answered by mr W last updated on 12/Jul/20

s a y \sqrt{x^{2} + y^{2}} = D

x^{2} + y^{2} = D^{2}

x^{2} + {(5 - \frac{5 x}{12})}^{2} = D^{2}

\frac{169}{144} x^{2} - \frac{50}{12} x + 25 - D^{2} = 0

Δ = {(\frac{50}{12})}^{2} - 4 \times \frac{169}{144} (25 - D^{2}) = 0

D^{2} = \frac{3600}{169}

\Rightarrow D = \frac{60}{13}

Answered by 1549442205 last updated on 13/Jul/20

Apply Cauchy - Shward we have

60^{2} = {(5 x + 12 y)}^{2} ⩽ (5^{2} + 12^{2}) (x^{2} + y^{2})

\Rightarrow x^{2} + y^{2} ⩾ \frac{60^{2}}{5^{2} + 12^{2}} = \frac{60^{2}}{13^{2}} \Rightarrow \sqrt{x^{2} + y^{2}} ⩾ \frac{60}{13}

the equality ocurrs if and only if

{\begin{cases} \frac{x}{5} = \frac{y}{12} \\ 5 x + 12 y = 60 \end{cases} \Leftrightarrow {\begin{cases} x = \frac{300}{169} \\ y = \frac{720}{169} \end{cases}

Thus, \sqrt{x^{2} + y^{2}} has the smallest value

equal \frac{60}{13} when (x, y) = (\frac{300}{169}; \frac{720}{169})