If-x-3-2-y-4-2-25-what-maximum-and-minimum-value-of-7x-8y-

Question Number 140270 by liberty last updated on 06/May/21

If {(x - 3)}^{2} + {(y - 4)}^{2} = 25,

what maximum and minimum

value of 7 x + 8 y ?

Answered by EDWIN88 last updated on 06/May/21

let {\begin{cases} x - 3 = 5 \cos θ \to x = 3 + 5 \cos θ \\ y - 4 = 5 \sin θ \to y = 4 + 5 \sin θ \end{cases}

let 7 (3 + 5 \cos θ) + 8 (4 + 5 \sin θ) = f (θ)

f (θ) = 35 \cos θ + 40 \sin θ + 53

f (θ) = \sqrt{35^{2} + 40^{2}} \cos (θ - α) + 53

where α = \tan^{- 1} (\frac{8}{7})

f (θ) = \sqrt{25 (49 + 64)} \cos (θ - α) + 53

{\begin{cases} f {(θ)}_{\max} = 5 \sqrt{113} + 53 \\ f {(θ)}_{\min} = - 5 \sqrt{113} + 53 \end{cases}

Answered by mr W last updated on 06/May/21

a n o t h e r w a y :

l e t 7 x + 8 y = k

\Rightarrow y = \frac{k - 7 x}{8}

{(x - 3)}^{2} + {(\frac{k - 7 x}{8} - 4)}^{2} = 25

113 x^{2} - (14 k - 64) x + k^{2} - 64 k = 0

s u c h t h a t x \in R e x i s t s,

Δ = {(14 k - 64)}^{2} - 4 \times 113 \times (k^{2} - 64 k) ⩾ 0

\Rightarrow k^{2} - 106 k - 16 ⩽ 0

k_{1, 2} = 53 \pm 5 \sqrt{113}

\Rightarrow 53 - 5 \sqrt{113} ⩽ k ⩽ 53 + 5 \sqrt{113}

i . e .

k_{m i n} = 53 - 5 \sqrt{113}

k_{m a x} = 53 + 5 \sqrt{113}

Facebook Tweet Pin