Ngā Rārangi Pānga ki ngā Wāhanga Kōnika
He ariā nui ngā wāhanga kōniko i roto i te pāngarau, inā koa i roto i te āhuahanga tātari. Ko te kupu "wāhanga kōniko" e pā ana ki te pihi i whiwhihia mā te whakawhitiwhitinga o te kōniko me te papa. E whā ngā momo matua o ngā wāhanga kōniko: te porowhita, te porowhita, te parabola, me te hyperbola. I roto i tēnei tuhinga, ka matapakihia e mātou te ariā o te pānihi ki te wāhanga kōniko me pēhea te whakamahi i tēnei ariā i roto i ngā āhuatanga rerekē.
Te Whakamāramatanga o te Raina Pānga
Ko te rārangi pātata he rārangi e pā ana ki tētahi piko i te pūwāhi kotahi anake, ā, kāore e whakawhiti i te piko i taua pūwāhi. I roto i te horopaki o ngā wāhanga kōniko, he maha ngā āhuatanga rerekē o ngā pātata i runga i te momo wāhanga kōniko e kōrerohia ana.
Pānga ki te Porowhita
He āhua motuhake te porowhita mō tētahi porowhita, he ōrite te roa o ngā tuaka matua e rua. Hei kimi i te pātata ki tētahi porowhita, ka whakamahia e mātou te whārite o te porowhita i te āhua paerewa:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
ko \((a, b)\) te pokapū o te porowhita, ā, ko \(r\) tōna radius.
Me kī tātou e hiahia ana kia mōhio ki te rārangi pātata i te pūwāhi \( (x_1, y_1) \). Ka taea te tuhi i te rārangi pātata i taua pūwāhi penei:
\[ (x – a)(x_1 – a) + (y – b)(y_1 – b) = r^2 \]
Rārangi Pānga ki te Porowhita
He wāhanga kōniko te porowhita e toro atu ana ki tētahi porowhita. Ko te whārite paerewa mō tētahi porowhita ko:
\[ \frac{(x – h)^2}{a^2} + \frac{(y – k)^2}{b^2} = 1 \]
ko \((h, k)\) te pokapū o te porowhita, ko \(a\) te tuaka haurua-matua, ā, ko \(b\) te tuaka haurua-iti.
Hei kimi i te rārangi pātapa i te pūwāhi \( (x_1, y_1) \) i runga i te porowhita, ka taea e tātou te whakamahi i te whārite e whai ake nei:
\[ \frac{(x_1 – h)(x – h)}{a^2} + \frac{(y_1 – k)(y – k)}{b^2} = 1 \]
He ahurei tēnei rārangi pātapa nā te mea kotahi anake te pūwāhi e pā ana ki te porowhita, ā, kāore e whakawhiti i te kōpiko.
Rārangi Pānga ki te Parabola
He wāhanga kōniko te parabola, kotahi te arotahi me te ahunga kotahi. Ko te whārite whānui o te parabola i te āhua paerewa ko:
\[ y^2 = 4ax \] rānei \[ x^2 = 4ay \]
Hei kimi i te rārangi pātata i te pūwāhi \( (x_1, y_1) \) i runga i te parabola \( y^2 = 4ax \), ka taea e tātou te whakamahi i te whārite:
\[ yy_1 = 2a(x + x_1) \]
He āhuatanga ahurei anō hoki tō te rārangi pātapa ki te parabola, arā, ka pā ki te piko i tētahi pūwāhi, engari kāore e whakawhitia.
Rārangi Pānga ki te Pūwero
He wāhanga kōniko te hyperbola, e rua ngā pihi tuwhera ōrite. Ko te whārite paerewa mō te hyperbola ko:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Hei kimi i te rārangi pātata i te pūwāhi \( (x_1, y_1) \) i runga i te hyperbola, ka whakamahia e mātou te whārite o te rārangi pātata:
\[ \frac{(x_1 – h)(x – h)}{a^2} – \frac{(y_1 – k)(y – k)}{b^2} = 1 \]
Ngā Taupānga Raina Pānga
He maha ngā whakamahinga o te ariā o ngā pātata ki ngā wāhanga kōniko i roto i te ao tūturu me te pūtaiao. Ko ētahi tauira:
1. Tirohanga Whānui: I roto i te hoahoa o ngā pūnaha tirohanga whāiti, pērā i ngā karu tiro me ngā karu hiko, he mea nui te mārama ki ngā pātapa ki ngā porowhita me ngā parabola hei arotahi i te mārama me te whakaiti i ngā hapa.
2. Te Aromatawai Tātai: He āhua porowhita te āhua o ngā ara o ngā aorangi me ngā amiorangi, nō reira, mā te mārama ki ngā tātai ka āwhina i te whakamahere i te ara nekehanga o ngā tinana o te rangi.
3. Hoahoa Whare me te Hangarau Ā-Iwi: Ko te hoahoa o ngā piriti, ngā tuanui, me ētahi atu hanganga he maha ngā wā ka whakamahia ngā āhua parabolic hei tohatoha kawenga tino pai.
4. Te Robotics me te Mātauranga Horihori: He maha ngā wā ka whakamahia e ngā rauropi mō te whakatere karetao me te mōhio ki ngā tauira ngā ariā āhuahanga pērā i te pātapa ki ngā wāhanga kōniko mō te whakamahere ara me te mōhio ki ngā mea.
5. Pāngarau me te Mātauranga: He tūāpapa nui te mārama ki te ariā o ngā pātapa ki ngā wāhanga kōniko i roto i te āhuahanga me te tātaitai, hei āwhina i ngā ākonga ki te whakawhanake i te māramatanga āhuahanga me ngā pūkenga tātari.
Tauira raruraru
Hei whakarato i tētahi pikitia whānui ake, me titiro tātou ki tētahi tauira o te whakamahi i tētahi rārangi pātapa ki tētahi parabola.
Pātai: Whakatauhia te whārite o te rārangi pātata ki te parabola \( y^2 = 8x \) e tika ana mā te pūwāhi \( (2, 4) \).
Whakautu:
I runga i te whārite o te parabola \( y^2 = 8x \) me te pūwāhi o te pātata \( (x_1, y_1) = (2, 4) \). Mā te whakamahi i te whārite o te rārangi pātata \( yy_1 = 2a(x + x_1) \), ka whakakapia e mātou \( a = 2 \) (nā te mea 4a = 8, nō reira a = 2), \( y_1 = 4 \), \( x_1 = 2 \):
\[ y \cdot 4 = 2 \cdot 2 \cdot (x + 2) \]
\[ 4y = 4(x + 2) \]
\[ y = x + 2 \]
Nō reira, ko te whārite o te rārangi pātata ki te parabola \( y^2 = 8x \) e tika ana mā te pūwāhi \( (2, 4) \) ko \( y = x + 2 \).
Whakamutunga
Kei roto i ngā wāhanga pātata ki ngā wāhanga kōniko ngā ariā me ngā tikanga maha mō te kimi rārangi e pā ana ki tētahi kōpiko i tētahi pūwāhi kotahi. He āwhina te mārama ki te mahi a ngā pātata ki ngā porowhita, ngā porowhita, ngā parabola, me ngā hyperbola i roto i ngā momo tono mahi me ngā tono mātauranga. Mā te māramatanga hohonu me te mahi auau, ka taea e ēnei ariā te noho hei taputapu tino whai hua i roto i ngā momo mara o te pūtaiao me te hangarau.
