Točno
23. svibnja 2012. 10:26 (12 godine, 1 mjesec)
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Zadatak smatram izrazito teškim za srednju školu jer nisam uspio naći rješenje koje ne zahtijeva poznavanje diferencijalnog računa.
Pramen pravaca kroz točku
određen je jednadžbom
, a sjecišta
zadane hiperbole i tih pravaca su rješenja sustava jednadžbi ![\begin{gather*}y = k\left(x - 2\right) + 5\\3x^2 - 4y^2 = 12 \text{.}\end{gather*}](/media/m/d/b/5/db50dcbbd98f8d3d545890d97aa7d6a0.png)
Jasno je da tražimo
za koji
postiže najmanju vrijednost. S obzirom da i
postiže najmanju vrijednost za isti
dovoljno je promatrati ![\begin{align*}{\left\vert PQ \right\vert}^2 &= \left(x_Q - 2\right)^2 + \left(y_Q - 5\right)^2\\ &= \left(x_Q - 2\right)^2 + k^2 \left(x_Q - 2\right)^2\\ &= \left(k^2 + 1\right) \left(x_Q - 2\right)^2 \text{.}\end{align*}](/media/m/0/3/4/034fddbc1abeb32f1a37ea47c4b5e398.png)
Rješavajući gore navedeni sustav jednadžbi dolazimo do![\left(3 - 4k^2\right)x_Q^2 + 8\left(2k - 5\right)kx_Q - 16\left(k^2 - 5k + 7\right) = 0 \text{.}](/media/m/e/4/1/e41d9933be8479ab8869b516e06fed17.png)
Za
, odnosno
, imamo
odakle je
, odnosno
ili
.
Za
dobijemo
pa je ![{\left\vert PQ \right\vert}^2 = \frac{k^2 + 1}{\left(3 - 4k^2\right)^2} \left(20k - 6 \pm 4\sqrt{3}\sqrt{7 - 5k}\right)^2 \text{.}](/media/m/0/0/2/002263e242d15eec8c7172c35415ac61.png)
U ovom trenutku možemo odabrati mukotrpno deriviranje složene funkcije ili pomoć nekog alata, npr. Wolfram Alphe.
Ove dvije funkcije postižu minimume
(za
) i približno
(za
):
1) http://www.wolframalpha.com/input/?i=minimum+%28k%5E2+%2B+1%29+%2F+%283+-+4k%5E2%29%5E2+*+%2820k+-+6+%2B+4Sqrt%5B3%5DSqrt%5B7+-+5k%5D%29%5E2
2) http://www.wolframalpha.com/input/?i=minimum+%28k%5E2+%2B+1%29+%2F+%283+-+4k%5E2%29%5E2+*+%2820k+-+6+-+4Sqrt%5B3%5DSqrt%5B7+-+5k%5D%29%5E2.
Sada je jasno da je točki
najbliža točka
(odgovara slučaju
) i da je njihova udaljenost
.
Pramen pravaca kroz točku
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![y = k\left(x - 2\right) + 5](/media/m/a/7/6/a76c7e9907b364044110ff8ab9a306f6.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
![\begin{gather*}y = k\left(x - 2\right) + 5\\3x^2 - 4y^2 = 12 \text{.}\end{gather*}](/media/m/d/b/5/db50dcbbd98f8d3d545890d97aa7d6a0.png)
Jasno je da tražimo
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![\left\vert PQ \right\vert](/media/m/4/e/b/4ebc54776fa4fb402f15b8dc5a775007.png)
![{\left\vert PQ \right\vert}^2](/media/m/8/5/a/85a3aed2acbb91b09a879b5cb51f7a16.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![\begin{align*}{\left\vert PQ \right\vert}^2 &= \left(x_Q - 2\right)^2 + \left(y_Q - 5\right)^2\\ &= \left(x_Q - 2\right)^2 + k^2 \left(x_Q - 2\right)^2\\ &= \left(k^2 + 1\right) \left(x_Q - 2\right)^2 \text{.}\end{align*}](/media/m/0/3/4/034fddbc1abeb32f1a37ea47c4b5e398.png)
Rješavajući gore navedeni sustav jednadžbi dolazimo do
![\left(3 - 4k^2\right)x_Q^2 + 8\left(2k - 5\right)kx_Q - 16\left(k^2 - 5k + 7\right) = 0 \text{.}](/media/m/e/4/1/e41d9933be8479ab8869b516e06fed17.png)
Za
![4k^2 = 3](/media/m/3/c/d/3cd21d9b2baceb85eb52857cd58f4f1f.png)
![k_{1,\,2} = \pm \dfrac{\sqrt{3}}{2}](/media/m/7/f/3/7f3fc44645b82d2967a5225cb67d8a2e.png)
![{x_Q}_{1,\,2} = \frac{57 \pm 125 \sqrt{3}}{66} \text{,}](/media/m/d/1/1/d11315c01f10c11a33ff780bd5f9bf92.png)
![{\left\vert PQ \right\vert}^2 = \dfrac{4375}{2904} \left(14 \pm 5\sqrt{3}\right)](/media/m/a/f/8/af8af5c243f2b1618e721deaec0672ac.png)
![{\left\vert PQ \right\vert}^2 \approx 8.04456](/media/m/e/3/b/e3bd97a1f6a86dd56d1a54bbf82732ca.png)
![{\left\vert PQ \right\vert}^2 \approx 34.13864](/media/m/5/8/d/58d29895b3cc03439796d06260468bc0.png)
Za
![4k^2 \neq 3](/media/m/4/b/f/4bf7b61630e75d91737964c9bdbfd79c.png)
![{x_Q}_{3,\,4} = \frac{-8k^2 + 20k \pm 4\sqrt{3}\sqrt{7 - 5k}}{3 - 4k^2}](/media/m/0/b/e/0be7a94fdab94e193b6918237efd6da5.png)
![{\left\vert PQ \right\vert}^2 = \frac{k^2 + 1}{\left(3 - 4k^2\right)^2} \left(20k - 6 \pm 4\sqrt{3}\sqrt{7 - 5k}\right)^2 \text{.}](/media/m/0/0/2/002263e242d15eec8c7172c35415ac61.png)
U ovom trenutku možemo odabrati mukotrpno deriviranje složene funkcije ili pomoć nekog alata, npr. Wolfram Alphe.
Ove dvije funkcije postižu minimume
![8](/media/m/3/d/2/3d2c45264dbff498f9bcb16af5f83881.png)
![k = -1](/media/m/c/a/6/ca6c00c361ae6e5cfe2441e864547da7.png)
![33.833](/media/m/c/c/5/cc564f26d8384ea10c670cbf0f784d80.png)
![k \approx 0.75475](/media/m/4/b/e/4be1ca95f40fc2180b49e4438cca363c.png)
1) http://www.wolframalpha.com/input/?i=minimum+%28k%5E2+%2B+1%29+%2F+%283+-+4k%5E2%29%5E2+*+%2820k+-+6+%2B+4Sqrt%5B3%5DSqrt%5B7+-+5k%5D%29%5E2
2) http://www.wolframalpha.com/input/?i=minimum+%28k%5E2+%2B+1%29+%2F+%283+-+4k%5E2%29%5E2+*+%2820k+-+6+-+4Sqrt%5B3%5DSqrt%5B7+-+5k%5D%29%5E2.
Sada je jasno da je točki
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![Q\!\left(4,\,3\right)](/media/m/8/2/6/826d1571baa5a7bed9804ecc380dd6bb.png)
![k = -1](/media/m/c/a/6/ca6c00c361ae6e5cfe2441e864547da7.png)
![2\sqrt{2}](/media/m/6/2/2/622955f13ad85d04cba4d59249a309ef.png)
Ocjene: (1)
Komentari:
abulj, 7. kolovoza 2012. 16:13
pbakic, 27. svibnja 2012. 13:53
U svakom slučaju, bez jačih tehnika ne ide? Da su brojevi drukčiji, možda bismo i uspjeli...
Ni meni nije uspjelo nešto lijepo... Probao sam s Lagrangeovim multiplikatorima (točno ovakvi zadaci dođu na kolokvijima na drugoj godini), ali opet ispadne dosta ružan sustav u kojem u najboljem slučaju pogađam rješenja kubne jednadžbe
Zadnja promjena: pbakic, 27. svibnja 2012. 13:54
ikicic, 23. svibnja 2012. 21:47