Mela e Tangent ho ea Likarolong tsa Conic
Likarolo tsa Conic ke mohopolo oa bohlokoa lipalo, haholo-holo ho jiometri ea tlhahlobo. Poleloana "karolo ea conic" e bolela mothinya o fumanoang ke ho kopana ha khoune le sefofane. Ho na le mefuta e mene e meholo ea likarolo tsa conic: selikalikoe, ellipse, parabola, le hyperbola. Sehloohong sena, re tla tšohla mohopolo oa tangent ho karolo ea conic le mokhoa oa ho sebelisa mohopolo ona maemong a fapaneng.
Tlhaloso ea Mohala oa Tangent
Mola o kobehileng ke mola o amang mothinya ntlheng e le 'ngoe feela 'me ha o kopane le mothinya ntlheng eo. Moelelong oa likarolo tsa konokono, li-tangent li na le litšobotsi tse 'maloa tse fapaneng ho latela mofuta oa karolo ea konokono e tšohloang.
Ho leba ho Selikalikoe
Selikalikoe ke mohlala o ikhethang oa ellipse eo ho eona li-axes tse peli tsa mantlha li lekanang ka bolelele. Ho fumana tangent ho selikalikoe, hangata re sebelisa equation ea selikalikoe ka mokhoa o tloaelehileng:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
moo \((a, b)\) e leng setsi sa selikalikoe 'me \(r\) ke radius ea sona.
A re re re batla ho tseba mola wa tangent ntlheng \( (x_1, y_1) \). Mola wa tangent ntlheng eo o ka ngolwa tjena:
\[ (x – a)(x_1 – a) + (y – b)(y_1 – b) = r^2 \]
Mohala o Tangent ho ea ho Ellipse
Ellipse ke karolo ea conic e leng katoloso ea selikalikoe. Tekanyo e tloaelehileng ea ellipse ke:
\[ \frac{(x – h)^2}{a^2} + \frac{(y – k)^2}{b^2} = 1 \]
moo \((h, k)\) e leng setsi sa ellipse, \(a\) ke semi-major axis, mme \(b\) ke semi-minor axis.
Ho fumana mola o tangent ntlheng \( (x_1, y_1) \) hodima ellipse, re ka sebedisa equation e latelang:
\[ \frac{(x_1 – h)(x – h)}{a^2} + \frac{(y_1 – k)(y – k)}{b^2} = 1 \]
Mola ona o motsu o ikhetha hobane o ama ellipse ntlheng e le 'ngoe feela 'me ha o kopane le mothinya.
Mohala o Mokhutšoanyane ho ea ho Parabola
Parabola ke karolo ea conic e nang le ntlha e le 'ngoe le directrix e le 'ngoe. Tekanyo e akaretsang ea parabola ka mokhoa o tloaelehileng ke:
\[ y^2 = 4ax \] kapa \[ x^2 = 4ay \]
Ho fumana mola o tangent ntlheng \( (x_1, y_1) \) ho parabola \( y^2 = 4ax \), re ka sebedisa equation:
\[ yy_1 = 2a(x + x_1) \]
Mola o motsu o yang ho parabola o boetse o na le thepa e ikgethang ya ho ama mothinya ntlheng e nngwe ntle le ho o kopanya.
Mohala o Mokhutšoanyane ho ea ho Hyperbola
Hyperbola ke karolo ea conic e nang le likhutlo tse peli tse bulehileng tse lekanang. Tekanyo e tloaelehileng ea hyperbola ke:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Ho fumana mola o tangent ntlheng \( (x_1, y_1) \) ho hyperbola, re sebedisa equation ya mola o tangent:
\[ \frac{(x_1 – h)(x – h)}{a^2} – \frac{(y_1 – k)(y – k)}{b^2} = 1 \]
Litšebeliso tsa Mohala oa Tangent
Khopolo ea li-tangent ho likarolo tsa conic e na le lits'ebetso tse fapaneng bophelong ba sebele le saenseng. Mehlala e meng ke ena:
1. Optics: Moralong oa litsamaiso tsa optical, tse kang litheleskoupo le li-microscopes, ho utloisisa li-tangent ho li-ellipses le li-parabolas ho bohlokoa bakeng sa ho tsepamisa leseli le ho fokotsa ho se lumellane.
2. Astronomica: Litsela tsa lipolanete le li-satellite hangata li latela sebopeho se kang elliptical, kahoo ho utloisisa li-tangent ho ka thusa ho rala tsela ea motsamao oa lihloliloeng tsa leholimo.
3. Meaho le Boenjiniere ba Meaho: Moralo oa marokho, mathule le meaho e meng hangata o sebelisa libopeho tsa papiso bakeng sa kabo e ntle ea mojaro.
4. Liroboto le Bohlale ba Maiketsetso: Li-algorithms tsa ho tsamaisa liroboto le temoho ea lipaterone hangata li sebelisa likhopolo tsa jeometri tse kang li-tangent ho likarolo tsa conic bakeng sa ho rala tsela le temoho ea lintho.
5. Lipalo le Thuto: Ho utloisisa mohopolo oa li-tangent ho likarolo tsa conic ke motheo oa bohlokoa oa jeometri le calculus, ho thusa baithuti ho nts'etsapele kutloisiso ea jeometri le tsebo ea tlhahlobo.
Mohlala oa mathata
Ho fana ka setšoantšo se felletseng haholoanyane, ha re shebeng mohlala oa ho sebelisa mola o nang le tangent ho parabola.
Potso: Fumana equation ea mola o tangent ho parabola \( y^2 = 8x \) e fetang ntlheng \( (2, 4) \).
Karabo:
Ka lebaka la equation ya parabola \( y^2 = 8x \) le ntlha ya tangency \( (x_1, y_1) = (2, 4) \). Re sebedisa equation ya mola wa tangent \( yy_1 = 2a(x + x_1) \), re nka sebaka sa \( a = 2 \) (hobane 4a = 8, kahoo a = 2), \( y_1 = 4 \), \( x_1 = 2 \):
\[ y \cdot 4 = 2 \cdot 2 \cdot (x + 2) \]
\[ 4y = 4(x + 2) \]
\[ y = x + 2 \]
Kahoo, equation ea mola o tangent ho parabola \( y^2 = 8x \) e fetang ntlheng \( (2, 4) \) ke \( y = x + 2 \).
Qetello
Di-tangents ho dikarolo tsa conic di akaretsa mehopolo le maqheka a fapaneng a ho fumana mela e amang sekhutlo se fanoeng ntlheng e le nngwe. Ho utlwisisa hore na di-tangents ho di-circles, di-ellipses, di-parabolas, le di-hyperbolas di sebetsa jwang ho ka thusa ditshebedisong tse fapaneng tsa tshebetso le tsa thuto. Ka kutlwisiso e phethahetseng le ho ikwetlisa kamehla, mehopolo ena e ka ba disebediswa tse thusang haholo mafapheng a fapaneng a saense le theknoloji.
