Kuenzana kweMutsetse weTangent kuDenderedzwa
Denderedzwa chimwe chezvinhu zvakakosha zvejometri uye chinowanzoonekwa munzvimbo dzakasiyana dzesainzi, kubva kumasvomhu ekutanga kusvika kumainjiniya ekuvaka nekuvaka. Imwe yepfungwa huru dzine chekuita nemadenderedzwa mu analytical geometry ndeye equation yemutsetse wetangent kune denderedzwa. Kunzwisisa equation yemutsetse wetangent kune denderedzwa kunovhura kunzwisisa kwakadzama kwehukama huripo pakati pezvinhu zvejometri uye mashandisirwo azvo muhupenyu hwezuva nezuva. Chinyorwa chino chichatsanangura equation yemutsetse wetangent kune denderedzwa zvakadzama, kutanga nepfungwa yekutanga uye kuwana equation, pamwe nekushandisa mienzaniso.
Pfungwa Yekutanga yeKutenderera kuDenderedzwa
Tangenti kudenderedzwa mutsetse unobata denderedzwa panzvimbo imwe chete usingaiparadzanise. Nzvimbo iyi apo mutsetse nedenderedzwa zvinosangana inonzi poindi yetangenti. Kusiyana nemitsetse inongosangana nedenderedzwa panzvimbo mbiri, tangenti dzine hunhu hwakasiyana hwekuti tangenti yega yega kudenderedzwa yakatarisana neradius yedenderedzwa panzvimbo iyoyo.
Maequations Akajairika eDenderedzwa neMitsetse
Usati wakurukura nezve equation yemutsetse wetangent, zvakakosha kutanga waziva equation yakajairika yedenderedzwa nemutsetse muCartesian coordinates.
Denderedzwa reEquation
Denderedzwa rine pakati panzvimbo \((h, k)\) uye radius \(r\) rine equation:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Equation yeMitsetse
Mitsetse iri muCartesian plane inogona kuratidzwa mumhando dzakasiyana-siyana, imwe yeinonyanya kuzivikanwa ndiyo slope-intercept form:
\[ y = mx + c \]
apo \(m\) iri gradient (kana slope) yemutsetse uye \(c\) iri intercept (cutter) iri pamusoro pe y-axis.
Kusarudza Equation yeTangent Line kuDenderedzwa
Kune nzira dzakasiyana siyana dzinogona kushandiswa kuona equation yemutsetse une tangent kune denderedzwa. Heano mamwe enzira dzakajairika.
Nzira 1: Kushandisa Mapoinzi eGradient neTangent
Kana tikaziva nzvimbo ine tangent \((x_1, y_1)\) padenderedzwa rine pakati \((h, k)\), tinogona kushandisa chimiro chejometri chekuti mutsetse une tangent wakatarisana ne radius yedenderedzwa pa tangent point. Kana gradient ye radius ichipfuura nepakati pema point \((h, k)\) uye \((x_1, y_1)\) iri:
\[ m_{radius} = \frac{y_1 – k}{x_1 – h} \]
Zvadaro gradient yemutsetse wetangent, uyo wakatarisana nemutsetse weradius, ndeiyi:
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{x_1 – h}{y_1 – k} \]
Kana tichiziva kuti mutsetse wetangent wakaita sei, tinogona kunyora equation yemutsetse wetangent muchimiro cheslope-intercept tichishandisa poindi \((x_1, y_1)\):
\[ y – y_1 = m_{tangent}(x – x_1) \]
Kana muchimiro chakajairika:
\[ y – y_1 = -\frac{x_1 – h}{y_1 – k}(x – x_1) \]
Nzira 2: Kushandisa Kutsiva uye Kusiyanisa
Kuti tiwane tangent kune denderedzwa rinozivikanwa tichishandisa nzira yekuchinja uye kusiyanisa, tinotanga nekunyora equation yedenderedzwa tobatanidza equation yakajairika yemutsetse. Equation yakajairika yemutsetse ndi \( y = mx + c \). Kubatanidza izvi ne equation yedenderedzwa:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Tsiva \( y \) mudenderedzwa re equation na \( mx + c \):
\[ (x – h)^2 + (mx + c – k)^2 = r^2 \]
Equation iyi inobva yawedzerwa kuita standard quadratic form \(Ax^2 + Bx + C = 0\). Kuti mutsetse uve wakabatana nedenderedzwa, panofanira kunge paine mhinduro imwe chete ye \(x\), saka discriminant ye quadratic equation inofanira kunge yakaenzana ne zero. Discriminant ye quadratic equation \(Ax^2 + Bx + C = 0\) ndeiyi:
\[ D = B^2 – 4AC \]
Ne \(D = 0\), tinogona kuona kukosha kwe \(m\) uye \(c\) kunoita kuti mutsetse utaridzike kune denderedzwa.
Mienzaniso yeKushandisa
Muenzaniso 1: Kusarudza Equation yeTangent Line
Ngatitii tine denderedzwa rine equation \( (x – 3)^2 + (y + 4)^2 = 25 \) uye tinoda kuziva equation yemutsetse wetangent unopfuura nepakati pepoindi \((-1, 5)\).
Kutanga, tinoongorora kana poindi iri padenderedzwa. Kuisa \((x, y) = (-1, 5)\) mu equation yedenderedzwa:
\[ (-1 – 3)^2 + (5 + 4)^2 = (-4)^2 + 9^2 = 16 + 81 = 97 \]
Sezvo \(97 \neq 25\), poindi iyi haisi padenderedzwa. Zvisinei, tinogona kuwana mutsetse unopfuura nepapoindi iyi uye wakatarisana neradius panzvimbo ye tangency.
Kutanga, tinowana gradient ye radius inopfuura nepakati pepoindi:
\[ m_{radius} = \frac{5 + 4}{-1 – 3} =\frac{9}{-4} = -\frac{9}{4} \]
Saka, gradient yemutsetse wetangent ndeiyi:
\[ m_{tangent} = -\frac{1}{m_{radius}} = \frac{4}{9} \]
Equation yemutsetse wetangent uchishandisa gradient iyi uye kupfuura nepakati pepoindi \((-1,5)\) ndeiyi:
\[ y – 5 = \frac{4}{9}(x + 1) \]
Mhedziso
Kuenzanisa kwe tangent nedenderedzwa ipfungwa huru yejometri, asi ine mashandisirwo akapararira munzvimbo dzakasiyana-siyana. Nekunzwisisa hunhu hwe tangents nenzira dzekuona ma equation avo, tinogona kushandisa pfungwa iyi kugadzirisa matambudziko akasiyana-siyana mumasvomhu nesainzi.
Kunzwisisa madenderedzwa nematanho kunovhurawo ruzivo rwakakura mukukura kwesainzi, kunyanya mumasvomhu ekuongorora. Kuburikidza nenzira yakarongeka, tinogona kubatanidza zvinhu zvakasiyana-siyana munzvimbo dzine mativi maviri, zvichisimbisa kunzwisisa kwedu kwezvinokosha zvejometri zvinogona kushanda sehwaro hwekutsvaga zvakanyanya mujometri nekuongorora nzvimbo.
