Muenzaniso weMibvunzo yeKukurukurirana yeMajiyomethri ekuongorora
Pendauluan
Analytical geometry ibazi remasvomhu rinosanganisa algebra negeometry kugadzirisa matambudziko ane chekuita nenzvimbo nechimiro. Ichi chishandiso chine simba chinotibvumira kuongorora matambudziko egeometry tichishandisa ma equation nema coordinates. Chinyorwa chino chichakurukura mienzaniso yakati wandei yematambudziko enzvimbo yegeometry yeanalytical uye tichakurukura zvakadzama kuti tinzwisise zvakadzama.
Muenzaniso Mubvunzo 1: Mutsetse weEquation
Mubvunzo:
Zvichipiwa mapoinzi maviri A(1, 2) naB(3, 7). Sarudza equation yemutsetse unopfuura nepakati pemapoinzi maviri aya.
Kukurukurirana:
Kuti tiwane equation yemutsetse unopfuura nepakati pemapoinzi maviri, tinogona kushandisa fomura ye gradient (slope) m:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Nepoindi A(x1, y1) = (1, 2) uye poindi B(x2, y2) = (3, 7):
\[ m = \frac{7 – 2}{3 – 1} = \frac{5}{2} \]
Tevere, tinoshandisa fomura yekuenzanisa kwemutsara:
\[ y – y_1 = m(x – x_1) \]
Kutsiva poindi imwe chete, semuenzaniso poindi A(1, 2):
\[ y – 2 = \frac{5}{2}(x – 1) \]
Chinja fomu iyi kuita equation yakajeka ye y:
\[ y – 2 = \frac{5}{2}x – \frac{5}{2} \]
\[ y = \frac{5}{2}x – \frac{5}{2} + 2 \]
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Saka, equation yemutsara ndeiyi:
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Muenzaniso Mubvunzo 2: Denderedzwa
Mubvunzo:
Sarudza equation yedenderedzwa rakanangana panzvimbo C(-2, 3) uye ine radius ye4.
Kukurukurirana:
Equation yakajairika yedenderedzwa rine pakati pa (h, k) uye radius r ndeiyi:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Kubva pamubvunzo, pakati pedenderedzwa (h, k) = (-2, 3) uye radius r = 4. Saka,
\[ (x + 2)^2 + (y – 3)^2 = 4^2 \]
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Saka, equation yedenderedzwa ndeiyi:
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Muenzaniso Mubvunzo 3: Parabola
Mubvunzo:
Sarudza equation ye parabola yakamira ine vertex pa (1, -2) uye focus pa (1, 0).
Kukurukurirana:
Pa parabola yakamira ine vertex (h, k), equation yakajairika ndeiyi:
\[(x – h)^2 = 4p(y – k) \]
Tichifunga nezve vertex (h, k) = (1, -2), tinofanira kuwana kukosha kwe p. Chinangwa che parabola ndi (h, k + p), uye kubva padambudziko chinangwa ndeche (1, 0):
\[k + p = 0 – (-2) = 2 \]
Kuti:
\[ p = 2 \]
Saka, equation yakajairika inova:
\[(x – 1)^2 = 4 \cdot 2 (y + 2) \]
\[(x – 1)^2 = 8(y + 2) \]
Saka, equation ye parabola ndeiyi:
\[(x – 1)^2 = 8(y + 2) \]
Muenzaniso Mubvunzo 4: Ellipse
Mubvunzo:
Kana wapiwa ellipse ine pakati panzvimbo (0, 0), major axis length 10 uye minor axis 6. Sarudza equation yeellipse.
Kukurukurirana:
Pakati pe ellipse (h, k) ndi (0, 0), kureba kwe major axis 2a = 10 zvekuti a = 5, uye kureba kwe minor axis 2b = 6 saka b = 3. Muedzo mukuru we ellipse ine centre pa (0, 0) ndewekuti:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Tsiva kukosha kwa a na b:
\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Saka, equation ye ellipse ndeiyi:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Muenzaniso Mubvunzo 5: Hyperbola
Mubvunzo:
Kana ukashandisa hyperbola ine pakati pa (1, -3), kureba kwe axis inochinjika i8 uye kureba kwe axis inochinjika i6. Sarudza equation ye hyperbola.
Kukurukurirana:
Kana paine hyperbola ine centre (h, k) uye horizontal transversal axis, general equation ndeiyi:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Pakati pe hyperbola (h, k) ndi (1, -3), kureba kwe transverse axis ndi 2a = 8 zvekuti a = 4, uye kureba kwe conjugate axis ndi 2b = 6 saka b = 3. Saka, equation ye hyperbola ndeiyi:
\[ \frac{(x – 1)^2}{4^2} – \frac{(y + 3)^2}{3^2} = 1 \]
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Saka, equation ye hyperbola ndeiyi:
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Mhedziso
Analytical geometry inzira ine simba yekuongorora maumbirwo ejometri uye maumbirwo uchishandisa algebraic equations. Nekunzwisisa pfungwa huru dze equations dzemitsara, madenderedzwa, parabolas, ellipses, uye hyperbolas, tinogona kugadzirisa zviri nyore matambudziko akasiyana-siyana ejometri. Chinyorwa chino chinopa mienzaniso nekukurukurirana kwezvinetso zvakakosha mu analytical geometry kuti zvikubatsire kudzikamisa kunzwisisa kwako. Mamwe matambudziko ekudzidzira anogona kubatsira kusimbisa nekuwedzera kunzwisisa kwako kwechinyorwa ichi.
