Mehlala ea Lipotso tse Buisanang ka Likarolo tsa Parabolic Conic
Karolo ea conic ke karolo ea bokaholimo ba conic bo khaotsoeng ke sefofane. Libopeho tsa jeometri tsa likarolo tsa conic li kenyelletsa li-circles, li-ellipses, li-parabola le li-hyperbola. Sehloohong sena, re tla tsepamisa maikutlo ho parabola, e leng e 'ngoe ea mefuta e tloaelehileng ea likarolo tsa conic tse kopanang le masimo a fapaneng a saense, haholo-holo lipalo le fisiks. Parabola e ka hlalosoa e le sete ea lintlha tse hole ka ho lekana le ntlha e tsitsitseng (sepheo) le mola o tsitsitseng (directrix).
Tlhaloso ea Parabola
Ho utloisisa haholoanyane mohopolo oa parabola, ho hlokahala ho utloisisa likarolo tse 'maloa tsa bohlokoa tsa parabola, e leng:
1. Ntlheng (Tlhōrō): Sebaka seo parabola e fetohang ho sona moo parabola e fetolang mothinya.
2. Tsepamiso: Ntlha e tsitsitseng ka hara sefofane e sebediswang ho hlalosa parabola.
3. Directrix: Mola o tsitsitseng ka hara sefofane o sebediswang ho hlalosa parabola.
4. Axis of Symmetry: Mola o fetang ka hara focus le vertex, mme o arola parabola ka dikarolo tse pedi tse lekanang.
Tekanyo e akaretsang ea parabola eo ntlha ea eona e leng qalong (0,0) e ka ngoloa ka mefuta e 'meli:
– Horizontal Parabola: \( y^2 = 4ax \)
– Parabola e otlolohileng: \( x^2 = 4ay \)
moo \(a\) e leng sebaka se tlohang ho vertex ho ya ho tsepamiso.
Lipotso le Lipuisano tsa Mehlala
Tse latelang ke mehlala e 'maloa ea lipotso le lipuisano tsa tsona tse amanang le parabolas.
Mohlala oa Potso ea 1
Potso:
Fumana equation ea parabola e nang le vertex qalong (0,0) le ntlha e tsepamisitsoeng ntlheng (3,0).
Puisano:
Ho tsoa potsong, re bona hore sepheo sa parabola se ntlheng ea (3,0). Kaha sepheo se ho x-axis e ntle, rea tseba hore parabola e tlameha ho ba e otlolohileng.
Bakeng sa parabola e rapameng, re sebedisa equation e akaretsang \( y^2 = 4ax \).
Kaha tlhokomelo e ho (3,0), joale \(a = 3\).
Kahoo, equation ea parabola ke:
\[ y^2 = 4 \cdot 3 \cdot x \]
\[ y^2 = 12x \]
Mohlala oa Potso ea 2
Potso:
Fumana equation ea parabola e nang le vertex qalong (0,0) le directrix x = -4.
Puisano:
Directrix ea parabola ke mola o tsitsitseng o hole ka ho fetisisa ho tloha vertex, o shebaneng le focus. Kahoo, haeba directrix e le x = -4, joale focus e ho (4,0).
Hape, sena se bontša hore parabola e rapame.
Sebaka se tlohang ho vertex ho ea ho tsepamiso, \(a = 4\).
Tekanyo ea parabola ke:
\[ y^2 = 4 \cdot 4 \cdot x \]
\[ y^2 = 16x \]
Mohlala oa Potso ea 3
Potso:
Ho fanoe ka parabola e nang le equation \( x^2 = 8y \). Fumana likhokahano tsa vertex, focus, le directrix equation.
Puisano:
Ho tsoa ho equation \(x^2 = 8y\), ho ka bonoa hore ena ke parabola e otlolohileng.
Bakeng sa parabola ea foromo \( x^2 = 4ay \), re ka bapisa:
\[ 4a = 8 \]
\[a = 2 \]
Sena se bontša hore sebaka ho tloha tlhorong ho ea sebakeng seo ho tsepamisitsoeng maikutlo ho sona ke 2.
– Dikoordinate tsa Tlhoro: Kaha ha ho na phetoho, tlhoro e dula e le moo e qalang teng (0, 0).
– Tsepamiso: Tsepamiso e hokahane le mothapo o motle wa y hole le ntlha ya a hole, e leng (0, 2).
– Directrix: Directrix ke mola wa y = -a, kahoo directrix ke y = -2.
Mohlala oa Potso ea 4
Potso:
Fumana equation ea parabola e nang le ntlha e tsepamisitsoeng (0, -2) le ntlha e ka holimo (0, 0).
Puisano:
Bothata bona bo bontša hore parabola e otlolohile ebile ea fokotseha (hobane sepheo se ka tlase ho vertex).
Bakeng sa parabola e otlolohileng e shebileng tlase, sebopeho se akaretsang ke \( x^2 = -4ay \).
Sebaka se tlohang ho vertex ho ea ho tsepamiso, \( a = 2 \).
Kahoo, equation ea parabola ke:
\[ x^2 = -4 \cdot 2 \cdot y \]
\[ x^2 = -8y \]
Mohlala oa Potso ea 5
Potso:
Parabola e na le equation \( y^2 + 4y – 4x + 20 = 0 \). Fumana dikhokahano tsa vertex ya yona, focus, le directrix.
Puisano:
Mohato oa 1: Fetola sebopeho sa equation ea parabola ho ea ho sebopeho se tloaelehileng.
Qala ka ho ngola equation hape:
\[ y^2 + 4y – 4x + 20 = 0 \]
\[ y^2 + 4y = 4x – 20 \]
Mohato oa 2: Qetella sekwere se phethahetseng bakeng sa karolo ea \(y\):
\[ y^2 + 4y + 4 = 4x – 20 + 4 \]
\[ (y + 2)^2 = 4x – 16 \]
\[ (y + 2)^2 = 4(x – 4) \]
Mohato oa 3: Bapisa le foromo e akaretsang \( (y – k)^2 = 4a(x – h) \). Tabeng ena, \(a = 1\), \(k = -2\), le \(h = 4\).
– Likhokahano tsa tlhoro: (4, -2)
– Tsepamiso: Kaha \(a = 1\), sebaka sa yona ho tloha ntlheng ke yuniti e le 1. Tsepamiso ke (4+1, -2) = (5, -2).
– Directrix: Mola o otlolohileng o feta ka \( x = h – a = 4 – 1 = 3 \). Kahoo, directrix ke \( x = 3 \).
Ka ho utloisisa mefuta e fapaneng ea mathata le mekhoa ea ona ea tharollo, kutloisiso ea hau ea li-parabola e tla ntlafala. Itloaetse mathata ka libopeho le libopeho tse fapaneng ho matlafatsa mohopolo ona. Li-parabola ha se mohopolo oa lipalo feela empa hape li na le lits'ebetso tse ngata fisiks le boenjiniere, ho kenyeletsoa le litsela tsa projectile le li-reflector tsa parabola litsamaisong tsa puisano. Ha u ntse u ikoetlisa haholoanyane, u tla tseba sehlooho sena haholoanyane.
