Isilinganiso Somugqa Oqondile Ku-Circle
Indilinga ingenye yezinto eziyisisekelo zejiyomethri futhi ivame ukuhlangana nayo emikhakheni eyahlukene yesayensi, kusukela kwizibalo eziyisisekelo kuya kubunjiniyela bezokwakha kanye nokwakhiwa kwezakhiwo. Omunye wemibono ebalulekile ehlobene nezindilinga ku-geometry yokuhlaziya yi-equation yomugqa we-tangent kuya ku-circle. Ukuqonda i-equation yomugqa we-tangent kuya ku-circle kuvula ukuqonda okujulile kobudlelwano phakathi kwezinto zejiyomethri kanye nokusetshenziswa kwazo ekuphileni kwansuku zonke. Lesi sihloko sizochaza i-equation yomugqa we-tangent kuya ku-circle ngokuningiliziwe, siqale ngomqondo oyisisekelo bese sithola i-equation, kanye nokusebenzisa izibonelo.
Umqondo Oyisisekelo Wokuthambekela Endilinga
I-tangent esindilinganeni umugqa othinta indilinga endaweni eyodwa kuphela ngaphandle kokuyihlanganisa. Leli phuzu lapho umugqa nesindilinga kuhlangana khona libizwa ngokuthi iphuzu le-tangency. Ngokungafani nemigqa ehlangana nje nesindilinga ezindaweni ezimbili, ama-tangent anesici esiyingqayizivele sokuthi yonke i-tangent esindilinga iqonde ngqo endaweni eqondile yesindilinga kuleyo ndawo.
Izilinganiso Ezijwayelekile Zemibuthano Nemigqa
Ngaphambi kokuxoxa ngesibalo somugqa we-tangent, kubalulekile ukwazi kuqala isibalo esijwayelekile sendilinga kanye nomugqa kuma-coordinates e-Cartesian.
Isilinganiso Sendilinga
Indilinga enesikhungo endaweni \((h, k)\) kanye nerediyasi \(r\) inesibalo:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Isibalo Somugqa
Imigqa endizeni yeCartesian ingavezwa ngezindlela eziningana, enye yazo evame kakhulu yifomu lokuvimba ukuthambeka:
\[ y = mx + c \]
lapho i-\(m\) iyi-gradient (noma i-slope) yomugqa kanye ne-\(c\) iyi-intercept (cutter) ezungeze i-y-axis.
Ukunquma Isilinganiso Somugqa Oqondile Kumbuthano
Kunezindlela eziningana ezingasetshenziswa ukunquma i-equation yomugqa oqondile nombuthano. Nazi ezinye zezindlela ezivame kakhulu.
Indlela 1: Ukusebenzisa Amaphuzu E-Gradient kanye ne-Tangent
Uma sazi iphuzu eliqondile \((x_1, y_1)\) kumbuthano onesikhungo \((h, k)\), singasebenzisa isici sejiyometri sokuthi umugqa oqondile uqonde ngqo kububanzi bendilinga endaweni eqondile. Uma i-gradient yerediyasi idlula kumaphuzu \((h, k)\) kanye \((x_1, y_1)\) ingu:
\[ m_{radius} = \frac{y_1 – k}{x_1 – h} \]
Khona-ke i-gradient yomugqa we-tangent, oqondile kumugqa we-radius, ithi:
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{x_1 – h}{y_1 – k} \]
Njengoba i-gradient yomugqa we-tangent yaziwa, singabe sesibhala i-equation yomugqa we-tangent ngesimo se-slope-intercept sisebenzisa iphuzu \((x_1, y_1)\):
\[ y – y_1 = m_{tangent}(x – x_1) \]
Noma ngesimo esijwayelekile:
\[ y – y_1 = -\frac{x_1 – h}{y_1 – k}(x – x_1) \]
Indlela 2: Ukusebenzisa Ukushintshana kanye Nokuhlukanisa
Ukuze sithole i-tangent kumbuthano owaziwayo sisebenzisa indlela yokufaka esikhundleni kanye nokuhlukanisa, siqala ngokubhala i-equation yembuthano bese sixhuma i-equation ejwayelekile yomugqa. I-equation ejwayelekile yomugqa ingu-\( y = mx + c \). Ukuhlanganisa lokhu ne-equation yembuthano:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Faka esikhundleni se-\( y \) ku-equation yesiyingi nge-\( mx + c \):
\[ (x – h)^2 + (mx + c – k)^2 = r^2 \]
Lesi sibalo sibe sesinwetshwa sibe yifomu elijwayelekile le-quadratic \(Ax^2 + Bx + C = 0\). Ukuze umugqa uhlangane nendilinga, kumele kube nesixazululo esisodwa ngqo se-\(x\), ngakho-ke i-discriminant ye-quadratic equation kumele ilingane no-zero. I-discriminant ye-quadratic equation \(Ax^2 + Bx + C = 0\) ithi:
\[ D = B^2 – 4AC \]
Nge-\(D = 0\), singakwazi ukuthola amanani ka-\(m\) kanye no-\(c\) enza umugqa uhlangane nendilinga.
Izibonelo Zokusebenza
Isibonelo 1: Ukunquma Isibalo Somugqa Oqondile
Ake sithi sinesiyingi esine-equation \( (x – 3)^2 + (y + 4)^2 = 25 \) futhi sifuna ukwazi i-equation yomugqa we-tangent odlula ephuzwini \((-1, 5)\).
Okokuqala, sihlola ukuthi iphuzu lisendilinga. Sifaka u-\((x, y) = (-1, 5)\) esilinganisweni sendilinga:
\[ (-1 – 3)^2 + (5 + 4)^2 = (-4)^2 + 9^2 = 16 + 81 = 97 \]
Kusukela ku-\(97 \neq 25\), leli phuzu alikho embuthanweni. Kodwa-ke, sisengathola umugqa odlula kuleli phuzu futhi oqonde ngqo ku-radius endaweni ye-tangency.
Okokuqala, sithola i-gradient ye-radius edlula ephuzwini:
\[ m_{radius} = \frac{5 + 4}{-1 – 3} =\frac{9}{-4} = -\frac{9}{4} \]
Ngakho-ke, i-gradient yomugqa we-tangent yile:
\[ m_{tangent} = -\frac{1}{m_{radius}} = \frac{4}{9} \]
Isibalo somugqa we-tangent osebenzisa le gradient kanye nokudlula ephuzwini \((-1,5)\) yilesi:
\[ y – 5 = \frac{4}{9}(x + 1) \]
Isiphetho
I-equation ye-tangent nendilinga ingumqondo oyisisekelo kakhulu wejometri, kodwa inezinhlelo zokusebenza ezibanzi emikhakheni ehlukahlukene. Ngokuqonda izakhiwo zama-tangent nezindlela zokunquma izilinganiso zawo, singasebenzisa lo mqondo ukuxazulula izinkinga ezahlukahlukene kwizibalo nesayensi.
Ukuqonda imibuthano kanye ne-tangents kuvula nokuqonda okubanzi ngentuthuko yesayensi, ikakhulukazi kwizibalo zokuhlaziya. Ngendlela ehlelekile, singaxhumanisa izinto ezahlukahlukene esikhaleni esinezinhlangothi ezimbili, siqinise ukuqonda kwethu izisekelo ze-geometric ezingasebenza njengesisekelo sokuhlola okwengeziwe kwi-geometry kanye nokuhlaziywa kwendawo.
