Su'aalo Tusaale ah iyo Doodo ku saabsan Goobo iyo Tangents
Wareegyada iyo dheellitirka waa laba mowduuc oo si joogto ah looga hadlo xisaabta, gaar ahaan heerka dugsiga sare. Fahmidda fikradda iyo adeegsiga dheellitirka wareegyada ayaa muhiim u ah qoto-dheeraynta aqoontaada joomatari. Maqaalkani wuxuu bixin doonaa tusaale ahaan dhibaatooyinka iyo doodaha ku saabsan wareegyada iyo dheellitirka si akhristayaasha loo siiyo faham qoto dheer.
Hordhac ku saabsan Aragtida Goobo iyo Tangents
Goobo
Goobo waa tiro dhibco ah oo ku jira diyaarad oo u dhow bar go'an oo loo yaqaan bartamaha goobada. Masaafadan go'an waxaa loo yaqaan gacanka goobada. Xisaab ahaan, goobada waxaa lagu qeexi karaa isla'egta:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
halkaas oo \((a, b)\) ay yihiin isku-duwayaasha bartamaha goobada iyo \(r\) ay tahay gacanka.
Tangent
Taangent-ka goobada waa xariiq taabanaysa goobada hal dhibic. Bartan waxaa loo yaqaan barta taangent-ka. Astaamaha ugu muhiimsan ee taangent-ku waa inuu ku toosan yahay gacanka laga soo qaaday bartamaha goobada ilaa barta taangent-ka.
Su'aalo iyo Doodo Tusaale ah
Su'aal 1aad: Go'aaminta Isle'egta Xariiqda Tangent
Su'aal:
Waxaa la siiyay goobaabin leh xarun ku taal \( (2, 3) \) iyo gacan 5. Go'aami isle'egta xariiqda tangent-ka ee goobada ku taal barta \( P \) oo leh isku-duwayaal \( (5, 7) \).
Dood:
Tallaabada 1: Hubi in barta \( P \) ay dhab ahaantii ku taal goobada.
Si aad u hubiso in \( P (5, 7) \) uu ku yaal goobaabin leh xarun \( (2, 3) \) iyo gacan \( 5 \), ku beddel isku-duwayaasha \( P \) isle'egta goobada:
\[ (x – 2)^2 + (y – 3)^2 = 5^2 \]
\[ (5 – 2)^2 + (7 – 3)^2 = 25 \]
\[ 3^2 + 4^2 = 25 \]
\[ 9 + 16 = 25 \]
Maadaama sinnaantu run tahay, barta \( P \) waxay ku taal goobada.
Tallaabada 2: Go'aami jihada wareegga ee dhex mara \( (2, 3) \) iyo \( (5, 7) \):
\[ m_{radius} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{7 – 3}{5 – 2} = \frac{4}{3} \]
Tallaabada 3: Jaangooynta xariiqda taangent-ka oo ku toosan jaangooynta gacanka (jaangooyada badeecaddu waa -1):
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]
Tallaabada 4: Go'aami isle'egta xariiqda tangent-ka adoo isticmaalaya barta \( P (5, 7) \):
\[ y – y_1 = m (x – x_1) \]
\[ y – 7 = -\frac{3}{4} (x – 5) \]
Fududee:
\[ y – 7 = -\frac{3}{4}x + \frac{15}{4} \]
\[ 4 sano – 28 = -3x + 15 \]
\[ 3x + 4y – 43 = 0 \]
Haddaba, isle'egta xariiqda tangent-ka waa:
\[ 3x + 4y – 43 = 0 \]
Su'aal 2: Go'aaminta Barta Isku-dhafka ee Isle'egta Xariiqda
Su'aal:
Waxaa la siiyay goobaabin leh isle'egta \( x^2 + y^2 = 25 \) iyo xariiq \( y = \frac{3}{4}x + 2 \). Go'aami barta isku-dhafka ah ee u dhaxaysa xariiqda iyo goobada.
Dood:
Tallaabada 1: Ku beddel isle'egta xariiqda isle'egta goobada:
Isle'egta goobada:
\[ x^2 + y^2 = 25 \]
Ku beddel \( y = \frac{3}{4}x + 2 \) isle'egta goobada:
\[ x^2 + \left(\frac{3}{4}x + 2\right)^2 = 25 \]
\[ x^2 + \left(\frac{9}{16}x^2 + \frac{12}{4}x + 4 \right) = 25 \]
\[ x^2 + \frac{9}{16}x^2 + \frac{6}{2}x + 4 = 25 \]
\[ x^2 + \frac{9}{16}x^2 + 3x + 4 = 25 \]
Tallaabada 2: Fududee isla'egta:
\[ 16x^2 + 9x^2 + 48x + 64 = 400 \]
\[ 25x^2 + 48x + 64 – 400 = 0 \]
\[ 25x^2 + 48x – 336 = 0 \]
Tallaabada 3: Helitaanka xididdada iyadoo la adeegsanayo qaacidada labajibbaaran:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
\[ a = 25, b = 48, c = -336 \]
\[ x = \frac{-48 \pm \sqrt{48^2 – 4 \cdot 25 \cdot (-336)}}{2 \cdot 25} \]
\[ x = \frac{-48 \pm \sqrt{2304 + 33600}}{50} \]
\[ x = \frac{-48 \pm \sqrt{35904}}{50} \]
\[ x = \frac{-48 \pm 189.501}{50} \]
Xulashada \( x \) ansax ah oo ku salaysan barta tangency (hal \( x \) oo keliya ayaa soo saari doonta dhibic tangency):
\[ x = \frac{141.501}{50} \qiyaastii 2.83 \]
\[ x \qiyaastii 2.83 \]
Tallaabada 4: Ku beddel \( x \) isle'egta xariiqda si aad u hesho \( y \):
\[ y = \frac{3}{4}(2.83) + 2 \]
\[ y \qiyaastii 2.12 + 2 \]
\[ y \qiyaastii 4.12 \]
Markaa, barta isku-dhafka u dhaxaysa xariiqda \( y = \frac{3}{4}x + 2 \) iyo goobada \( x^2 + y^2 = 25 \) waa \( (2.83, 4.12) \).
Gabagabo
Barashada fikradaha wareegyada iyo dheellitirka waxaa ka mid ah fahamka aasaaska joomatari iyo awoodda lagu xallin karo dhibaatooyinka iyadoo la adeegsanayo isle'egyada xisaabta. Dhibaatooyinka sida kuwa kor ku xusan waxay ardayda ka caawiyaan inay ku dhaqmaan adeegsiga aragtida xaaladaha dhabta ah. Iyadoo la adeegsanayo ku celcelin joogto ah, ardayda waxaa laga filayaa inay si fudud u fahmaan oo ay u xalliyaan dhibaatooyinka.
