Isibonelo Semibuzo Yengxoxo Yejiyomethri Yokuhlaziya
I-Pendahuluan
I-analytical geometry igatsha lezibalo elihlanganisa i-algebra kanye ne-geometry ukuxazulula izinkinga ezihilela isikhala nesimo. Iyithuluzi elinamandla elisivumela ukuthi sihlaziye izinkinga ze-geometry sisebenzisa ama-equation nama-coordinates. Lesi sihloko sizoxoxa ngezibonelo eziningana zezinkinga zendawo ye-analytical geometry futhi sixoxe ngazo ngokuningiliziwe ukuze kube lula ukuqonda okujulile.
Isibonelo Umbuzo 1: Isibalo Somugqa
Umbuzo:
Uma unikezwe amaphuzu amabili u-A(1, 2) no-B(3, 7). Thola isibalo somugqa odlula kula maphuzu amabili.
Ingxoxo:
Ukuze sithole i-equation yomugqa odlula emaphuzwini amabili, singasebenzisa ifomula ye-gradient (slope) m:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Ngephuzu A(x1, y1) = (1, 2) kanye nephuzu B(x2, y2) = (3, 7):
\[ m = \frac{7 – 2}{3 – 1} = \frac{5}{2} \]
Okulandelayo, sisebenzisa ifomula yesibalo somugqa:
\[ y – y_1 = m(x – x_1) \]
Ukushintshaniswa kwephuzu elilodwa, isibonelo iphuzu A(1, 2):
\[ y – 2 = \frac{5}{2}(x – 1) \]
Guqula leli fomu libe yi-equation ecacile ka-y:
\[ y – 2 = \frac{5}{2}x – \frac{5}{2} \]
\[ y = \frac{5}{2}x – \frac{5}{2} + 2 \]
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Ngakho-ke, isibalo somugqa sithi:
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Isibonelo Umbuzo 2: Indilinga
Umbuzo:
Thola i-equation yendilinga ephakathi nendawo ku-C(-2, 3) kanye ne-radius engu-4.
Ingxoxo:
Isibalo esijwayelekile sendilinga enesikhungo ku-(h, k) kanye ne-radius r yilesi:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Kusukela embuzweni, isikhungo sendilinga (h, k) = (-2, 3) kanye nerediyasi r = 4. Ngakho-ke,
\[ (x + 2)^2 + (y – 3)^2 = 4^2 \]
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Ngakho-ke, isibalo sendilinga sithi:
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Isibonelo Umbuzo 3: I-Parabola
Umbuzo:
Nquma isibalo sepharabola eqondile ene-vertex ku-(1, -2) bese ugxila ku-(1, 0).
Ingxoxo:
Ku-parabola eqondile ene-vertex (h, k), i-equation ejwayelekile yile:
\[ (x – h)^2 = 4p(y – k) \]
Njengoba sinikezwe i-vertex (h, k) = (1, -2), sidinga ukuthola inani lika-p. Ukugxila kwe-parabola kungu-(h, k + p), futhi kusukela enkingeni ukugxila kungu-(1, 0):
\[k + p = 0 – (-2) = 2 \]
Ukuze:
\[p = 2 \]
Ngakho-ke, i-equation ejwayelekile iba:
\[ (x – 1)^2 = 4 \cdot 2 (y + 2) \]
\[ (x – 1)^2 = 8(y + 2) \]
Ngakho-ke, i-equation ye-parabola ithi:
\[ (x – 1)^2 = 8(y + 2) \]
Isibonelo Umbuzo 4: I-Ellipse
Umbuzo:
Uma unikezwe i-ellipse enesikhungo endaweni (0, 0), ubude be-axis enkulu engu-10 kanye ne-axis encane engu-6. Thola i-equation ye-ellipse.
Ingxoxo:
Isikhungo se-ellipse (h, k) singu-(0, 0), ubude be-axis enkulu 2a = 10 ukuze u-a = 5, kanye nobude be-axis encane 2b = 6 ukuze u-b = 3. Isibalo esijwayelekile se-ellipse esinesikhungo ku-(0, 0) singu-(0, 0) singu-(10).
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Faka amanani ka-a no-b esikhundleni salokho:
\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Ngakho-ke, i-equation ye-ellipse yile:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Isibonelo Umbuzo 5: I-Hyperbola
Umbuzo:
Uma unikezwe i-hyperbola enesikhungo ku-(1, -3), ubude be-axis ephambeneyo bungu-8 kanti ubude be-axis ehlanganisiwe bungu-6. Thola isibalo se-hyperbola.
Ingxoxo:
Ku-hyperbola ene-centre (h, k) kanye ne-horizontal transversal axis, i-equation ejwayelekile yile:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Isikhungo se-hyperbola (h, k) singu-(1, -3), ubude be-axis ephambeneyo bungu-2a = 8 ukuze u-a = 4, kanti ubude be-axis ehlanganisiwe bungu-2b = 6 ukuze u-b = 3. Ngakho-ke, isibalo se-hyperbola singu-:
\[ \frac{(x – 1)^2}{4^2} – \frac{(y + 3)^2}{3^2} = 1 \]
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Ngakho-ke, i-equation ye-hyperbola ithi:
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Isiphetho
I-geometry yokuhlaziya iyindlela enamandla yokuhlaziya izimo nezakhiwo zejiyometri kusetshenziswa ama-equation e-algebraic. Ngokuqonda imiqondo eyisisekelo yama-equation emigqa, imibuthano, ama-parabola, ama-ellipses, nama-hyperbola, singaxazulula kalula izinkinga ezahlukahlukene zejiyometri. Lesi sihloko sinikeza izibonelo nezingxoxo zezinkinga ezibalulekile ku-geometry yokuhlaziya ukusiza ukujulisa ukuqonda kwakho. Izinkinga ezengeziwe zokuzijwayeza zingasiza ekuqiniseni nasekukhuliseni ukuqonda kwakho kwalokhu okuqukethwe.
