Tekanyo ea Mohala oa Tangent ho Selikalikoe
Selikalikoe ke e 'ngoe ea lintho tsa bohlokoa ka ho fetisisa tsa jeometri 'me hangata e fumanoa mafapheng a fapaneng a saense, ho tloha lipalo tsa motheo ho ea boenjiniere ba sechaba le meaho. E 'ngoe ea likhopolo tsa bohlokoa tse amanang le lisekoere ho jiometri ea tlhahlobo ke equation ea mola o tangent ho ea ho selikalikoe. Ho utloisisa equation ea mola o tangent ho ea ho selikalikoe ho bula kutloisiso e tebileng ea likamano lipakeng tsa lintho tsa jeometri le ts'ebeliso ea tsona bophelong ba letsatsi le letsatsi. Sengoloa sena se tla hlalosa equation ea mola o tangent ho ea ho selikalikoe ka botlalo, ho qala ka mohopolo oa motheo le ho fumana equation, hammoho le ho sebelisa mehlala.
Khopolo ea Motheo ea Tangent ho Selikalikoe
Tangent ho selikalikoe ke mola o amang selikalikoe ntlheng e le 'ngoe feela ntle le ho se kopanya. Ntlha ena eo mola le selikalikoe li kopanang ho eona e bitsoa ntlha ea tangency. Ho fapana le mela e kopanang selikalikoe lintlheng tse peli feela, tangent e na le thepa e ikhethang ea hore tangent e 'ngoe le e 'ngoe ho selikalikoe e otlolohile ho radius ea selikalikoe ntlheng eo.
Litekanyo tse Akaretsang tsa Li-circles le Mela
Pele o buisana ka equation ea mola o nang le tangent, ho hlokahala hore o tsebe pele equation e akaretsang ea selikalikoe le mola ka har'a li-coordinate tsa Cartesian.
Tekanyo ea Selikalikoe
Selikalikoe se nang le bohareng ntlheng ea (h, k) le radius ea (r) se na le equation:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Tekanyo ea Mola
Mela e karolong ea Cartesian e ka hlahisoa ka mefuta e 'maloa, e' ngoe ea tse tloaelehileng haholo ke mokhoa oa ho thibela leralla:
\[ y = mx + c \]
moo \(m\) e leng gradient (kapa leralla) la mola mme \(c\) e le intercept (seha) e mabapi le y-axis.
Ho Fumana Tekanyo ea Mola o Tangent ho Selikalikoe
Ho na le mekhoa e 'maloa e ka sebelisoang ho fumana equation ea mola o tangent ho selikalikoe. Mona ke tse ling tsa mekhoa e tloaelehileng haholo.
Mokhoa oa 1: Ho Sebelisa Lintlha tsa Gradient le Tangent
Haeba re tseba ntlha e otlolohileng \((x_1, y_1)\) hodima selikalikoe se nang le setsi \((h, k)\), re ka sebedisa thepa ya jeometri eo mola o otlolohileng ho radius ya selikalikoe ntlheng e otlolohileng. Haeba gradient ya radius e fetang dintlheng \((h, k)\) le \((x_1, y_1)\) e le:
\[ m_{radius} = \frac{y_1 – k}{x_1 – h} \]
Ebe gradient ea mola o otlolohileng, o otlolohileng ho mola oa radius, ke:
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{x_1 – h}{y_1 – k} \]
Ka ho tseba gradient ea mola o tangent, re ka ngola equation ea mola o tangent ka mokhoa oa ho thibela leralla re sebelisa ntlha \((x_1, y_1)\):
\[ y – y_1 = m_{tangent}(x – x_1) \]
Kapa ka mokhoa o tloaelehileng:
\[ y – y_1 = -\frac{x_1 – h}{y_1 – k}(x – x_1) \]
Mokhoa oa 2: Ho Sebelisa Phetolo le Khethollo
Ho fumana tangent ho selikalikoe se tsejoang re sebelisa mokhoa oa ho nkela sebaka le oa ho khetholla, re qala ka ho ngola equation ea selikalikoe ebe re kenya equation e akaretsang ea mola. Equation e akaretsang ea mola ke \( y = mx + c \). Ho kopanya sena le equation ea selikalikoe:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Nka sebaka sa \( y \) ho equation ya sedikadikwe ka \( mx + c \):
\[ (x – h)^2 + (mx + c – k)^2 = r^2 \]
Tekanyo ena e atolosoa ho ba sebopeho se tloaelehileng sa quadratic \(Ax^2 + Bx + C = 0\). Hore mola o shebane le selikalikoe, ho tlameha ho be le tharollo e le 'ngoe hantle bakeng sa \(x\), kahoo karohano ea equation ea quadratic e tlameha ho lekana le lefela. Khethollo ea equation ea quadratic \(Ax^2 + Bx + C = 0\) ke:
\[ D = B^2 – 4AC \]
Ka \(D = 0\), re ka fumana boleng ba \(m\) le \(c\) bo etsang hore mola o kopane le selikalikoe.
Mehlala ea Ts'ebeliso
Mohlala oa 1: Ho Fumana Tekanyo ea Mola oa Tangent
A re re re na le selikalikoe se nang le equation \( (x – 3)^2 + (y + 4)^2 = 25 \) mme re batla ho tseba equation ya mola o tangent o fetang ntlheng \((-1, 5)\).
Taba ea pele, re hlahloba hore na ntlha e selikalikoe. Ho kenya \((x, y) = (-1, 5)\) sebakeng sa equation ea selikalikoe:
\[ (-1 – 3)^2 + (5 + 4)^2 = (-4)^2 + 9^2 = 16 + 81 = 97 \]
Kaha \(97 \neq 25\), ntlha ena ha e sedikeng. Leha ho le jwalo, re ntse re ka fumana mola o fetang ntlheng ena mme o otlolohile ho radius ntlheng ya tangency.
Taba ea pele, re fumana gradient ea radius e fetang ntlheng:
\[ m_{radius} = \frac{5 + 4}{-1 – 3} =\frac{9}{-4} = -\frac{9}{4} \]
Kahoo, gradient ea mola oa tangent ke:
\[ m_{tangent} = -\frac{1}{m_{radius}} = \frac{4}{9} \]
Tekanyo ea mola o tangent o sebelisang gradient ena le ho feta ntlheng \((-1,5)\) ke:
\[ y – 5 = \frac{4}{9}(x + 1) \]
Qetello
Tekanyo ea tangent ho selikalikoe ke mohopolo oa motheo oa jeometri, leha ho le joalo e na le lits'ebetso tse pharaletseng masimong a fapaneng. Ka ho utloisisa litšobotsi tsa tangent le mekhoa ea ho fumana li-equation tsa tsona, re ka sebelisa mohopolo ona ho rarolla mathata a fapaneng a lipalo le saense.
Ho utloisisa didikadikwe le di-tangent ho boetse ho bula temohisiso e batsi mabapi le ntshetsopele ya saense, haholoholo dipalong tsa tlhahlobo. Ka mokgwa o hlophisehileng, re ka hokahanya dikarolo tse fapaneng sebakeng sa mahlakore a mabedi, ra matlafatsa kutlwisiso ya rona ya metheo ya jeometri e ka sebetsang e le motheo wa ho fuputsa ho eketsehileng jeometri le tlhahlobo ya sebaka.
