Autor Tema: Skup tacaka  (Pročitano 1787 puta)

yahtzee

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Skup tacaka
« poslato: Oktobar 10, 2013, 08:02:12 posle podne »
Ako moze da mi neko rijesi ovaj zadatak:
Naci skup tacaka koji zadovoljavaju uslov Im(Z-Z1/Z-Z2)=0
Hvala.
“Ah you think darkness is your ally? You merely adopted the dark. I was born in it, molded by it. I didn't see the light until I was already a man, by then it was nothing to me but blinding!”

DeuS

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Odg: Skup tacaka
« Odgovor #1 poslato: Oktobar 10, 2013, 09:08:12 posle podne »
Naci skup tacaka koji zadovoljavaju uslov Im(Z-Z1/Z-Z2)=0

z = x + iy

(Z-Z1)/(Z-Z2) = (x + iy - x1 - iy1)/(x + iy - x2 - iy2) = [x - x1 + i(y- y1)]/[x - x2 + i(y - y2)]

Sad racionalizujemo. Pomnozimo sa konjugovano-kompleksnim parom kompleksnog broja u imeniocu. U imeniocu cemo dobiti realan broj, a u brojiocu cemo dobiti izraz:

[x - x1 + i(y- y1)]â‹…[x - x2 - i(y - y2)]

Kad to izmnozimo, dobicemo:

(neki realan deo) + i[(x - x2)(y- y1)-(x - x1)(y - y2)]

Po uslovu zadatka, ovo sto je u uglastoj zagradi (imaginarni deo) treba da bude 0.

(x - x2)(y- y1)-(x - x1)(y - y2) = 0

xy - x2y - xy1 + x2y1 - xy + x1y + xy2 - x1y2 = 0

Kad se to sracuna, dobije se

(y2 - y1)x + (x1 - x2)y = x1y2 - x2y1

tj.

(x1 - x2)y = (y1 - y2)x + x1y2 - x2y1

y = [(y1 - y2)x + x1y2 - x2y1]/(x1 - x2)

A to odgovara linearnoj funkciji.
We've had to endure much, you and I, but soon there will be order again. A new age. Aquinas spoke of the mythical city on the hill, soon that city will be real and we will be crowned its kings. Or, better than kings... Gods!