Tusaale Su'aalaha Doodda Joomatari ee Falanqaynta
Pendahuluan
Joomatari falanqayneed waa laan xisaabeed oo isku daraysa aljabrada iyo joomatari si loo xalliyo masalooyinka ku lug leh booska iyo qaabka. Waa qalab awood leh oo noo oggolaanaya inaan falanqeyno masalooyinka joomatari iyadoo la adeegsanayo isla'egyada iyo isku-duwayaasha. Maqaalkani wuxuu ka hadli doonaa dhowr tusaale oo ku saabsan masalooyinka aagga joomatari falanqayneed wuxuuna si faahfaahsan uga hadli doonaa si loo fududeeyo faham qoto dheer.
Su'aal Tusaale 1: Isle'egta Sadarka
Su'aal:
Laba dhibcood oo A(1, 2) iyo B(3, 7) la bixiyay. Go'aami isle'egta xariiqda dhex marta labada dhibcood.
Dood:
Si loo helo isle'egta xariiq dhex marta laba dhibcood, waxaan isticmaali karnaa qaacidada jihada (jiirada) m:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Iyada oo leh dhibic A(x1, y1) = (1, 2) iyo dhibic B(x2, y2) = (3, 7):
\[ m = \frac{7 – 2}{3 – 1} = \frac{5}{2} \]
Marka xigta, waxaan u isticmaalnaa qaacidada isle'egta xariiqda:
\[ y – y_1 = m(x – x_1) \]
Beddelka hal dhibic, tusaale ahaan dhibicda A(1, 2):
\[ y – 2 = \frac{5}{2}(x – 1) \]
Foomkan u beddel isla'eg cad oo loogu talagalay y:
\[ y – 2 = \frac{5}{2}x – \frac{5}{2} \]
\[ y = \frac{5}{2}x – \frac{5}{2} + 2 \]
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Haddaba, isle'egta xariiqda waa:
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Tusaale Su'aal 2aad: Goobo
Su'aal:
Go'aami isle'egta goobada oo ku taal barta C(-2, 3) oo leh gacan 4 ah.
Dood:
Isle'egta guud ee goobada oo leh xarun ku taal (h, k) iyo radius r waa:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Su'aasha laga soo bilaabo, bartamaha goobada (h, k) = (-2, 3) iyo gacanka r = 4. Markaa,
\[ (x + 2)^2 + (y – 3)^2 = 4^2 \]
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Haddaba, isle'egta goobada waa:
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Su'aal Tusaale ah 3: Parabola
Su'aal:
Go'aami isle'egta parabola toosan oo leh vertex at (1, -2) oo diiradda saar (1, 0).
Dood:
Parabola toosan oo leh lafo (h, k), isle'egta guud waa:
\[ (x – h)^2 = 4p(y – k) \]
Marka la eego geeska (h, k) = (1, -2), waxaan u baahanahay inaan helno qiimaha p. Diiradda parabola waa (h, k + p), dhibaatadana diiradda waa (1, 0):
\[k + p = 0 – (-2) = 2 \]
Sidaas darteed:
\[ p = 2 \]
Markaa, isla'egta guud waxay noqonaysaa:
\[ (x – 1)^2 = 4 \cdot 2 (y + 2) \]
\[ (x – 1)^2 = 8(y + 2) \]
Haddaba, isle'egta parabola waa:
\[ (x – 1)^2 = 8(y + 2) \]
Su'aal Tusaale 4: Ellipse
Su'aal:
Marka la eego ellipse oo leh xarun barta (0, 0), dhererka dhidibka weyn 10 iyo dhidibka yar 6. Go'aami isle'egta ellipse-ka.
Dood:
Bartamaha ellipse-ka (h, k) waa (0, 0), dhererka dhidibka weyn 2a = 10 sidaas darteed a = 5, iyo dhererka dhidibka yar 2b = 6 sidaas darteed b = 3. Isle'egta guud ee ellipse-ka oo leh xarun ku taal (0, 0) waa:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Beddel qiimaha a iyo b:
\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Markaa, nooca ellipse-ka waa sidan soo socota:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Tusaale Su'aal 5aad: Hyperbola
Su'aal:
Marka la eego hyperbola oo leh xarun ku taal (1, -3), dhererka dhidibka isdhaafka ah waa 8 dhererka dhidibka isdhaafka ahna waa 6. Go'aami isle'egta hyperbola.
Dood:
Hyperbola oo leh dhidibka dhexe (h, k) iyo dhidibka wareega ee toosan, isla'egta guud waa:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Xarunta hyperbola-da (h, k) waa (1, -3), dhererka dhidibka isdhaafka ah waa 2a = 8 si a = 4, dhererka dhidibka isdhaafka ahna waa 2b = 6 si b = 3. Sidaa darteed, isle'egta hyperbola-da waa:
\[ \frac{(x – 1)^2}{4^2} – \frac{(y + 3)^2}{3^2} = 1 \]
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Haddaba, isle'egta hyperbola waa:
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Gabagabo
Joomatari falanqayneed waa hab awood badan oo lagu falanqeeyo qaababka joomatari iyo qaab-dhismeedka iyadoo la adeegsanayo isla'egyada aljabrada. Annagoo fahmayna fikradaha aasaasiga ah ee isle'egyada xariiqyada, wareegyada, parabolas, ellipses, iyo hyperbolas, si fudud ayaan u xallin karnaa dhibaatooyin joomatari oo kala duwan. Maqaalkani wuxuu bixiyaa tusaalooyin iyo doodo ku saabsan dhibaatooyin muhiim ah oo ku jira joomatari falanqayneed si uu kaaga caawiyo inaad si qoto dheer u fahamto. Dhibaatooyinka tababarka dheeraadka ah waxay kaa caawin karaan xoojinta iyo ballaarinta fahamkaaga ku saabsan agabkan.
