Ihe atụ nke Ajụjụ Mkparịta ụka Jiometri Nyocha
Pendahuluan
Usoro nyocha bụ ngalaba mgbakọ na mwepụ nke na-ejikọta algebra na geometry iji dozie nsogbu ndị metụtara oghere na ọdịdị. Ọ bụ ngwaọrụ dị ike nke na-enye anyị ohere inyocha nsogbu geometry site na iji nha nha na nhazi. Isiokwu a ga-atụle ọtụtụ ihe atụ nke nsogbu mpaghara geometry nyocha ma tụlee ha nke ọma iji mee ka nghọta miri emi dị.
Ajụjụ Ihe Nlereanya nke 1: Nha nhata ahịrị
Ajụjụ:
E nyere isi ihe abụọ A(1, 2) na B(3, 7). Chọpụta nha nhata nke ahịrị gafere isi ihe abụọ a.
Azịza:
Iji chọta nha anya nke ahịrị gafere isi ihe abụọ, anyị nwere ike iji usoro gradient (m) mee ihe:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Site na isi A(x1, y1) = (1, 2) na isi B(x2, y2) = (3, 7):
\[ m = \frac{7 – 2}{3 – 1} = \frac{5}{2} \]
Na-esote, anyị na-eji usoro maka nha nhata nke ahịrị ahụ:
\[ y – y_1 = m(x – x_1) \]
Ndochi otu isi, dịka ọmụmaatụ isi A(1, 2):
\[ y – 2 = \frac{5}{2}(x – 1) \]
Gbanwee ụdị a ka ọ bụrụ usoro nhazi doro anya maka y:
\[ y – 2 = \frac{5}{2}x – \frac{5}{2} \]
\[ y = \frac{5}{2}x – \frac{5}{2} + 2 \]
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Ya mere, usoro nke ahịrị ahụ bụ:
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Ihe atụ Ajụjụ nke Abụọ: Okirikiri
Ajụjụ:
Chọpụta nha anya nke okirikiri dị n'etiti isi ihe C(-2, 3) na radius nke 4.
Azịza:
Usoro nhazi izugbe nke okirikiri nwere etiti na (h, k) na radius r bụ:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Site na ajụjụ a, etiti okirikiri ahụ (h, k) = (-2, 3) na radius r = 4. Ya mere,
\[ (x + 2)^2 + (y – 3)^2 = 4^2 \]
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Ya mere, usoro nke okirikiri ahụ bụ:
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Ajụjụ Ihe atụ nke 3: Parabola
Ajụjụ:
Chọpụta nha anya nke parabola kwụ ọtọ na vertex na (1, -2) ma lekwasị anya na (1, 0).
Azịza:
Maka parabola kwụ ọtọ nwere vertex (h, k), nhazi izugbe bụ:
\[ (x – h)^2 = 4p(y – k) \]
Ebe ọ bụ na vertex (h, k) = (1, -2), anyị kwesịrị ịchọta uru nke p. Isi ihe parabola bụ (h, k + p), site na nsogbu ahụ, isi ihe a bụ (1, 0):
\[k + p = 0 – (-2) = 2 \]
Ka ọ were:
\[ p = 2 \]
Ya mere, usoro izugbe ahụ ga-abụ:
\[ (x – 1)^2 = 4 \cdot 2 (y + 2) \]
\[ (x – 1)^2 = 8(y + 2) \]
Ya mere, usoro nke parabola bụ:
\[ (x – 1)^2 = 8(y + 2) \]
Ajụjụ Ihe Nlereanya nke 4: Ellipse
Ajụjụ:
E nyere ellipse nwere etiti na isi (0, 0), ogologo axis ukwu 10 na obere axis 6. Chọpụta nha nha nke ellipse ahụ.
Azịza:
Etiti nke ellipse (h, k) bụ (0, 0), ogologo nke isi axis 2a = 10 nke mere na a = 5, na ogologo nke obere axis 2b = 6 nke mere na b = 3. Usoro izugbe maka ellipse nwere etiti na (0, 0) bụ:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Dochie ụkpụrụ nke a na b:
\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Yabụ, usoro nke ellipse bụ:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Ajụjụ Ihe atụ nke 5: Hyperbola
Ajụjụ:
Ọ bụrụ na e jiri hyperbola nke nwere etiti dị na (1, -3), ogologo nke axis transverse bụ 8 na ogologo nke axis conjugate bụ 6. Chọpụta nha anya nke hyperbola ahụ.
Azịza:
Maka hyperbola nwere etiti (h, k) na axis transversal kwụ ọtọ, nhazi izugbe bụ:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Ebe etiti hyperbola (h, k) bụ (1, -3), ogologo nke axis transverse bụ 2a = 8 nke mere na a = 4, na ogologo nke axis conjugate bụ 2b = 6 nke mere na b = 3. Ya mere, nha anya nke hyperbola bụ:
\[ \frac{(x – 1)^2}{4^2} – \frac{(y + 3)^2}{3^2} = 1 \]
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Ya mere, usoro nke hyperbola bụ:
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Mmechi
Usoro nyocha bụ ụzọ dị ike maka inyocha ọdịdị na nhazi nke geometric site na iji usoro algebra. Site n'ịghọta echiche ndị bụ isi nke usoro nha anya nke ahịrị, okirikiri, parabolas, ellipses, na hyperbolas, anyị nwere ike idozi ọtụtụ nsogbu geometry ngwa ngwa. Isiokwu a na-enye ihe atụ na mkparịta ụka nke nsogbu dị mkpa na usoro nyocha iji nyere aka mee ka nghọta gị dịkwuo omimi. Nsogbu omume ndị ọzọ nwere ike inye aka mee ka nghọta gị nke ihe a sikwuo ike ma gbasaa.
