Zitsanzo za Mafunso Okambirana za Ma Parabolic Conic
Gawo la conic ndi gawo la pamwamba pa conic lodulidwa ndi ndege. Mawonekedwe a geometrical a magawo a conic akuphatikizapo mabwalo, ma ellipses, ma parabola, ndi ma hyperbola. M'nkhaniyi, tiyang'ana kwambiri pa parabola, imodzi mwa mitundu yodziwika bwino ya magawo a conic omwe amapezeka m'magawo osiyanasiyana a sayansi, makamaka masamu ndi fizikisi. Parabola ikhoza kutanthauziridwa ngati gulu la mfundo zomwe zili kutali kwambiri ndi mfundo yokhazikika (chidwi) ndi mzere wokhazikika (directrix).
Tanthauzo la Parabola
Kuti mumvetse bwino lingaliro la parabola, ndikofunikira kumvetsetsa zinthu zingapo zofunika za parabola, zomwe ndi:
1. Vertex (Nsonga): Malo osinthira parabola pomwe parabola imasintha ma curve.
2. Kuyang'ana: Malo okhazikika omwe amagwiritsidwa ntchito pofotokoza parabola.
3. Directrix: Mzere wokhazikika womwe umagwiritsidwa ntchito pofotokoza parabola.
4. Mzere wa Symmetry: Mzere womwe umadutsa mu focus ndi vertex, ndikugawa parabola m'magawo awiri ofanana.
Chiyerekezo cha parabola chomwe vertex yake ili pachiyambi (0,0) chingalembedwe m'njira ziwiri:
- Parabola Yopingasa: \( y^2 = 4ax \)
– Parabola Yoyimirira: \( x^2 = 4ay \)
kumene \(a\) ndi mtunda wochokera pa vertex kupita ku focus.
Mafunso ndi Zokambirana za Zitsanzo
Nazi zitsanzo zingapo za mafunso ndi zokambirana zawo zokhudzana ndi ma parabola.
Chitsanzo cha Funso 1
Funso:
Dziwani equation ya parabola yomwe ili ndi vertex koyambira (0,0) ndi focus pamalo (3,0).
Kukambirana:
Kuchokera mu funsoli, tikuona kuti cholinga cha parabola chili pa mfundo (3,0). Popeza cholinga chake chili pa x-axis yabwino, tikudziwa kuti parabola iyenera kukhala yopingasa.
Pa parabola yopingasa, timagwiritsa ntchito equation yonse \( y^2 = 4ax \).
Popeza cholinga chachikulu chili pa (3,0), ndiye \(a = 3\).
Kotero, equation ya parabola ndi:
\[ y^2 = 4 \cdot 3 \cdot x \]
\[ y^2 = 12x \]
Chitsanzo cha Funso 2
Funso:
Dziwani equation ya parabola yomwe ili ndi vertex koyambira (0,0) ndi directrix x = -4.
Kukambirana:
Directrix ya parabola ndi mzere wokhazikika womwe uli kutali kwambiri ndi vertex, moyang'anizana ndi focus. Chifukwa chake, ngati directrix ndi x = -4, ndiye kuti focus ili pa (4,0).
Apanso, izi zikusonyeza kuti parabola ndi yopingasa.
Mtunda kuchokera pa vertex kupita ku focus, \(a = 4\).
Chiyerekezo cha parabola ndi:
\[ y^2 = 4 \cdot 4 \cdot x \]
\[ y^2 = 16x \]
Chitsanzo cha Funso 3
Funso:
Kupatsidwa parabola yokhala ndi equation \( x^2 = 8y \). Dziwani ma coordinates a vertex, focus, ndi directrix equation.
Kukambirana:
Kuchokera ku equation \(x^2 = 8y\), zitha kuwoneka kuti iyi ndi parabola yoyimirira.
Kuti tipeze parabola ya mawonekedwe \( x^2 = 4ay \), tikhoza kuyerekeza:
\[ 4a = 8 \]
\[a = 2 \]
Izi zikusonyeza kuti mtunda wochokera pamwamba mpaka pamalo ofunikira ndi 2.
– Ma Coordinates a Peak: Popeza palibe kusintha, peak imakhalabe komwe idayambira (0, 0).
– Kuyang'ana Kwambiri: Kuyang'ana kwambiri kuli pa positive y-axis patali ndi a kuchokera pa vertex, yomwe ndi (0, 2).
– Directrix: Directrix ndi mzere wa y = -a, kotero directrix ndi y = -2.
Chitsanzo cha Funso 4
Funso:
Dziwani equation ya parabola yomwe ili ndi focus pa point (0, -2) ndi vertex pa point (0, 0).
Kukambirana:
Vutoli likuwonetsa kuti parabola ndi yoyima komanso yotsika (chifukwa cholinga chake chili pansi pa vertex).
Pa parabola yoyimirira yoyang'ana pansi, mawonekedwe onse ndi \( x^2 = -4ay \).
Mtunda kuchokera pa vertex kupita ku focus, \( a = 2 \).
Kotero, equation ya parabola ndi:
\[ x^2 = -4 \cdot 2 \cdot y \]
\[ x^2 = -8y \]
Chitsanzo cha Funso 5
Funso:
Parabola ili ndi equation \( y^2 + 4y – 4x + 20 = 0 \). Dziwani ma coordinates a vertex, focus, ndi directrix yake.
Kukambirana:
Gawo 1: Sinthani mawonekedwe a parabola equation kukhala mawonekedwe wamba.
Yambani polembanso equation:
\[ y^2 + 4y – 4x + 20 = 0 \]
\[ y^2 + 4y = 4x – 20 \]
Gawo 2: Malizitsani sikweya yoyenera ya gawo la \(y\):
\[ y^2 + 4y + 4 = 4x – 20 + 4 \]
\[ (y + 2)^2 = 4x – 16 \]
\[ (y + 2)^2 = 4(x – 4) \]
Gawo 3: Yerekezerani ndi mawonekedwe onse \( (y – k)^2 = 4a(x – h) \). Pankhaniyi, \(a = 1\), \(k = -2\), ndi \(h = 4\).
– Ma Coordinates a Nsonga: (4, -2)
– Kuyang'ana Kwambiri: Popeza \(a = 1\), mtunda wake kuchokera ku vertex ndi 1 unit. Kuyang'ana kwakukulu ndi (4+1, -2) = (5, -2).
– Directrix: Mzere woyima umadutsa mu \( x = h – a = 4 – 1 = 3 \). Chifukwa chake, directrix ndi \( x = 3 \).
Mwa kumvetsetsa mitundu yosiyanasiyana ya mavuto ndi njira zawo zothetsera mavuto, kumvetsetsa kwanu kwa ma parabola kudzasintha. Yesetsani mavuto okhala ndi mawonekedwe ndi mawonekedwe osiyanasiyana kuti mulimbikitse lingaliro ili. Ma parabola si lingaliro la masamu lokha komanso ali ndi ntchito zambiri mu fizikisi ndi uinjiniya, kuphatikiza njira zoyendetsera zinthu ndi zowunikira za parabola m'makina olumikizirana. Mukamachita zambiri, mudzadziwa bwino nkhaniyi.
