Su'aalo Tusaale ah iyo Ka Doodda Qaybaha Konik-ka ee Tangent-ka
Pendahuluan
Qaybta koonka waa qalooc ka dhasha isgoyska diyaarad leh koonka dipolar. Qalloocyadan waxaa ka mid ah goobo, ellipses, parabolas, iyo hyperbolas. Mawduuc muhiim ah oo ku saabsan fahamka qaybaha koonka waa xariiqda tangent. Taangent-ka qaybta koonka waa xariiq taabanaysa qalooca koonka hal dhibic oo keliya. Maqaalkani wuxuu ka hadli doonaa dhowr dhibaato oo tusaale ah iyo dood ku saabsan taangent-ka qaybaha koonka.
Taabasho u leh Goobo
Goobo waa qayb koon ah oo leh qaabka ugu fudud iyo isku-dheelitir qumman. Aan ku bilowno tusaale dhibaato ku saabsan tangents-ka iyo goobo.
Su'aal Tusaale 1aad
La siiyay goobaabin leh isla'egta \( (x – 2)^2 + (y + 3)^2 = 25 \). Go'aami isle'egta xariiqda tangent ee barta \((5, -3)\) ee goobada.
Dood
Isle'egta guud ee goobada waa \( (x – h)^2 + (y – k)^2 = r^2 \), iyadoo \( (h, k) \) ay tahay bartamaha goobada iyo \( r \) ay tahay gacanka. Dhibaatadan, bartamaha goobada \((h, k)\) waa \((2, -3)\) iyo gacanka \( r = \sqrt{25} = 5 \).
Xariiqda tangent-ka ee barta \((x_1, y_1)\) ee goobada waxaa laga heli karaa iyadoo la isticmaalayo qaacidada soo socota:
\[ (x – h)(x_1 – h) + (y – k)(y_1 – k) = r^2 \]
Geli qiimayaasha la yaqaan:
\[ (x - 2) (5 - 2) + (y + 3) (-3 + 3) = 25 \]
\[ (x - 2) (3) + (y + 3) (0) = 25 \]
\[ 3(x – 2) = 25 \]
\[ 3x – 6 = 25 \]
\[ 3x = 31 \]
\[ x = \frac{31}{3} \]
Isle'egta xariiqda tangent-ka waa \(x = \frac{31}{3}\), laakiin waxaa jira qalad ku jira habkan sababtoo ah barta \((5, -3)\) si cad ayay u tahay dhibic ku taal goobada. Sidaa darteed, waxaan isticmaalnaa habka dhaqameedka annagoo ku beddeleyna jiirada xariiqda tangent-ka bartan gaarka ah:
Barta tayjantiga waa, \((5, -3)\), markaa, tayjantiga (m) ee xariiqda radius waa \(m = \frac{-3 – (-3)}{5 – 2}=0\), halkaas oo tayjantiga xariiqda tayjantiga uu noqdo mid aan la qeexin tangent toosan ee ka horreysay.
Khadadka Tangent ilaa Ellipse
Ellipse waa qayb koon ah oo leh laba dhidib oo isku-dhafan: dhidib weyn (dheer) iyo dhidib yar (gaaban). Waa kuwan tusaalooyin qaar oo ah dhibaatooyinka ellipses.
Su'aal Tusaale 2aad
Marka la eego ellipse oo leh isla'egta \(\frac{x^2}{16} + \frac{y^2}{9} = 1\). Go'aami isle'egta xariiqda taangent ee barta \((2, \frac{3}{2})\) ee ku taal ellipse-ka.
Dood
Isle'egta xariiqda tangent ee ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) barta \((x_1, y_1)\) waa:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Iyada oo la adeegsanayo \(a = 4\) iyo \(b = 3\), beddel qiimayaasha \(a\), \(b\), iyo dhibicda \(2, \frac{3}{2})\):
\[ \frac{x(2)}{4^2} + \frac{y(\frac{3}{2})}{3^2} = 1 \]
\[ \frac{2x}{16} + \frac{3y}{6} = 1 \]
\[ \frac{x}{8} + \frac{y}{2} = 1 \]
Ku dhufo isla'egta oo dhan 8 si aad u tirtirto jajabyada:
\[ x + 4y = 8 \]
Markaa, isle'egta xariiqda tangent ee ellipse waa \( x + 4y = 8 \).
Khadka Tangent ilaa Parabola
Parabola waa qayb koon ah oo leh hal dhidib oo isku dheelitiran iyo hal geesood. Waa kuwan tusaalooyin qaar oo ah dhibaatooyinka la xiriira parabolas.
Su'aal Tusaale 3aad
Marka la eego parabola oo leh isla'egta \( y^2 = 4x \). Go'aami isle'egta xariiqda tangent ee barta \((1, 2)\) ee parabola.
Dood
Isle'egta xariiqda tangent ee parabola \( y^2 = 4ax \) barta \((x_1, y_1)\) waa:
\[ yy_1 = 2a(x + x_1) \]
Laga soo bilaabo isla'egta parabola \( y^2 = 4x \), waxaan ka helnaa \( 4a = 4 \) si \( a = 1 \). Ku beddel qiimaha \( a \) iyo dhibicda \((1, 2)\):
\[ 2y = 2(1)(x + 1) \]
\[ 2y = 2x + 2 \]
\[ y = x + 1 \]
Markaa, isle'egta xariiqda tangent ee parabola waa \( y = x + 1 \).
Khadka Tangent ilaa Hyperbola
Hyperbola waa qayb koon ah oo leh laba laamood iyo laba calaamadood oo asymptotes ah. Waa kuwan tusaalooyin dhibaatooyin ah oo la xiriira hyperbolas.
Su'aal Tusaale 4aad
Marka la eego hyperbola oo leh isla'egta \( \frac{x^2}{25} – \frac{y^2}{16} = 1 \). Go'aami isle'egta xariiqda tangent ee barta \((5, 0)\) ee hyperbola.
Dood
Isle'egta xariiqda tangent ee hyperbola \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\) barta \((x_1, y_1)\) waa:
\[ \frac{xx_1}{a^2} – \frac{yy_1}{b^2} = 1 \]
Iyada oo la adeegsanayo \(a = 5 \) iyo \(b = 4 \), beddel qiimayaasha \(a \), \(b \), iyo dhibicda \(5, 0)\):
\[ \frac{x(5)}{25} – \frac{y(0)}{16} = 1 \]
\[ \frac{5x}{25} – 0 = 1 \]
\[ \frac{x}{5} = 1 \]
\[ x = 5 \]
Markaa, isle'egta xariiqda tangent ee hyperbola waa \( x = 5 \).
Gabagabo
Tangents-ka qaybaha konikku waxay door muhiim ah ka ciyaaraan xisaabta iyo codsiyada kala duwan ee wax ku oolka ah. Fahmidda sida loo helo isle'egyada tangents-ka noocyada kala duwan ee qaybaha konikka, sida goobo, ellipses, parabolas, iyo hyperbolas, waa xirfad lagama maarmaan u ah xisaabinta iyo joomatari falanqaynta. Iyada oo la adeegsanayo tusaalooyinka iyo doodaha kor ku xusan, waxaa la rajeynayaa in akhristayaashu ay si fiican u fahmi doonaan fikradaha iyo hababka lagu go'aaminayo tangents-ka qaybaha konikka.
