Te Whārite Porowhita i roto i te Āhuahanga
He mea nui te kōpiko porowhita i roto i te āhuahanga, e puta ana i roto i ngā horopaki maha, mai i te pāngarau parakore ki ngā tono i roto i te ahupūngao, te hangarau, me te whetū. Ki te kī ngāwari, ka taea te mārama ki te porowhita hei "porowhita kua totorohia" kia roa ake ai i te taha kotahi. Heoi, he mea tino whakamere ake te whakamāramatanga ōkawa o te porowhita: ko te porowhita te huinga o ngā pūwāhi katoa i roto i te papa e pumau tonu ana te tapeke o ō rātou tawhiti mai i ngā pūwāhi pumau e rua (e kiia nei ko ngā arotahi). Mai i tēnei whakamāramatanga, ka taea te tango me te ako i te whārite o te porowhita, i roto i ngā āhua paerewa me ngā āhua whānui.
1. Te Mārama ki ngā Porowhita me ō rātou Huānga
Hei mārama ki te whārite o te porowhita, me mōhio tātou ki ngā āhuatanga matua o te porowhita:
1. Te pokapū o te porowhita (pokapū): te pūwāhi o te porowhita, e tohuhia ana ko \((h, k)\).
2. Tuaka matua: te diameter roa rawa atu o te porowhita.
3. Tuaka iti: te diameter poto rawa o te porowhita e tū poutū ana ki te tuaka matua.
4. Arotahi (foci): e rua ngā pūwāhi pumau hei tohutoro mō te whakamāramatanga o te porowhita, e tohuhia ana ko \(F_1\) me \(F_2\).
5. Pūtoro haurua-nui: te haurua o te roa o te tuaka matua, e tohuhia ana ko \(a\).
6. Te pūtoro haurua-iti: te haurua o te roa o te tuaka iti, e tohuhia ana ko \(b\).
7. Te tawhiti mai i te pokapū ki te arotahi: kua tohua ko \(c\), me te whanaungatanga porowhita noa:
\[
c^2 = a^2 – b^2
\]
He maha ngā wā ka puta he taupatupatu ariā i konei: i roto i te porowhita, ka mau tonu te \(a \ge b\) ā, kei runga i te tuaka matua ngā arotahi.
Hei tāpiritanga, kei reira te ariā o te āhua rerekē \(e\) e ine ana i te "whakataha whakarunga" o te porowhita:
\[
e = \frac{c}{a}, \quad 0 \le e < 1
\]
Jika \(e = 0\), elips menjadi lingkaran (karena \(c = 0\), fokus berimpit di pusat).
2. Persamaan Standar Elips Berpusat di Titik Asal
Jika elips berpusat di titik asal \((0,0)\) dan sumbu-sumbunya sejajar sumbu koordinat, persamaan elips memiliki bentuk standar yang sangat dikenal.
a) Sumbu mayor horizontal
Jika sumbu mayor sejajar sumbu-\(x\), maka:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
dengan \(a > b\). Fokus terletak pada sumbu-\(x\), yaitu di titik:
\[
(\pm c, 0), \quad \text{me } c^2 = a^2 – b^2
\]
b) Tuaka matua poutū
Mena he whakarara te tuaka matua ki te tuaka-y, kāti:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
me te \(a > b\). Ko te arotahi kei te tuaka-\(y\), arā:
\[
(0, \pm c), \quad c^2 = a^2 – b^2
\]
Mā tēnei āhua paerewa ka māmā ake te pānui i ngā āhuatanga o te porowhita: ko ngā uara o \(a\) me \(b\) e tohu tika ana i te rahi o te porowhita, ko \(c\) ia e whakatau ana i te tūranga o ngā arotahi.
3. Te Whārite Porowhita i te Pokapū o \((h,k)\)
I roto i te maha o ngā raruraru āhuahanga tātari, kāore te porowhita e pokapū tonu ana i te pokapū taunga. Mena kei te pokapū te porowhita i \((h,k)\), ka huri te whārite paerewa ki:
a) Tuaka matua whakapae
\[
\frac{(xh)^2}{a^2} + \frac{(yk)^2}{b^2} = 1
\]
b) Tuaka matua poutū
\[
\frac{(xh)^2}{b^2} + \frac{(yk)^2}{a^2} = 1
\]
He nekehanga (whakawhitinga) noa iho tēnei panonitanga o te porowhita, i te tīmatanga i waenganui i te pūtake. Ka nekehia hoki te arotahi ki te pokapū hou:
– Mō te tuaka matua whakapae: \((h \pm c, k)\)
– Mō te tuaka matua poutū: \((h, k \pm c)\)
4. Mai i te Whakamāramatanga o te Arotahi ki te Whārite o te Porowhita
Ka taea te whakamahi i te whakamāramatanga o te porowhita hei te tapeke o ngā tawhiti ki ngā arotahi pūmau e rua hei pūtake mō te whakaputa whārite. Hei tauira, me kī kei \((c,0)\) me \((-c,0)\ ngā arotahi, ā, ko te pūwāhi i runga i te porowhita ko \((x,y)\). Ko ngā tawhiti o taua pūwāhi ki ia arotahi ko:
\[
d_1 = \sqrt{(xc)^2 + y^2}, \quad d_2 = \sqrt{(x+c)^2 + y^2}
\]
Nā te mea he pumau te rahinga:
\[
d_1 + d_2 = 2a
\]
Mā te whakahaere i te taurangi (te tapawhā rua hei tango i ngā pūtake), ka whiwhi tātou i te whārite:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
me te \(b^2 = a^2 – c^2\). E whakaatu ana tēnei ehara te āhua paerewa o te porowhita i te tātai "kua maumaharatia" noa iho, engari i ahu mai i tētahi whakamāramatanga ā-ira.
5. Te Whārite Whānui o te Porowhita me tōna Tautuhinga
I roto i te mahi, he maha ngā wā ka tūtaki tātou ki ngā whārite tapawhā me ngā taurangi e rua kāore i te āhua paerewa, hei tauira:
\[
Ax^2 + By^2 + Cx + Dy + E = 0
\]
Ka taea e tētahi whārite pēnei te whakaatu i tētahi porowhita, i tētahi parabola, i tētahi hyperbola rānei. Hei whakarite kia rite ki tētahi porowhita (me ngā tuaka e whakarara ana ki ngā taunga), ko te tikanga me rite a \(A\) me \(B\):
– tohu kotahi (e rua ngā tohu pai, e rua rānei ngā tohu kino),
– ā, kāore i te rite te rahi i te nuinga o te wā (mēnā he rite te rahi, ā, kāore he kupu \(xy\), he tino tūponotanga he porowhita te āhua).
Hei huri ki te āhua porowhita paerewa, ko te tikanga e whakamahia whānuitia ana ko te whakaoti i te tapawhā i runga i ngā kupu \(x\) me \(y\). He tauira māmā noa iho:
\[
4x^2 + 9y^2 – 8x + 18y – 5 = 0
\]
Rōpū:
\[
4(x^2 – 2x) + 9(y^2 + 2y) = 5
\]
Whakaotia te tapawhā:
\[
4[(x-1)^2 – 1] + 9[(y+1)^2 – 1] = 5
\]
\[
4(x-1)^2 + 9(y+1)^2 = 5 + 4 + 9 = 18
\]
Mō te 18:
\[
\frac{(x-1)^2}{\frac{18}{4}} + \frac{(y+1)^2}{2} = 1
\]
koinei te āhua paerewa o te porowhita me te pokapū \((1,-1)\).
6. Ngā Whakamahinga o ngā Porowhita i roto i te Āhuahanga me te Ao Tūturu
Ehara i te mea he mea ariā noa iho ngā porowhita. I roto i te āhuahanga me te pūtaiao tono, he mahi nui tā ngā porowhita:
1. Te Tātaritanga whetū (Te Ture a Kepler): he porowhita te porowhita o te aorangi, ko te Rā kei te arotahi kotahi.
2. Ngā tirohanga me ngā oro: ko te āhuatanga o te whakaata porowhita e kī ana ka whakaatahia ngā ngaru mai i tētahi arotahi mā roto i tētahi atu arotahi. Ka whakamahia tēnei i roto i te hoahoa o ngā whare kōnohete, o ētahi whakaata whakaata rānei.
3. Te hangarau mīhini: ka whakamahia e ētahi tikanga taputapu, tikanga kāmera rānei ngā ara porowhita.
4. Hoahoa: mā te āhua porowhita ka whakakotahi i te ataahua me te mahi oro.
Mā te mārama ki te whārite porowhita, ka taea e tātou te tātari i te rahi, te tūranga, me ngā āhuatanga o ngā ara i roto i ngā pūnaha rerekē.
7. Whakamutunga
Mā te whārite o te porowhita i roto i te āhuahanga ka honoa te āputa i waenga i te whakamāramatanga āhuahanga (te tapeke o ngā tawhiti ki ngā arotahi pūmau e rua) me te whakaaturanga tātari (he whārite taurangi i roto i ngā taunga). Mā te āhua paerewa o te porowhita ka māmā ake te tautuhi i te pokapū, te roa o ngā tuaka, me ngā tūranga o ngā arotahi, engari ka taea te huri i ngā āhua whānui ki te āhua paerewa mā te whakaoti i te tapawhā. Mā te mārama ki ngā porowhita ka āwhina i te whakaoti rapanga āhuahanga tātari engari ka whakatuwhera hoki i ngā māramatanga ki te whakamārama a te pāngarau i ngā āhuatanga taiao pēnei i ngā porowhita aorangi me ngā āhuatanga o te whakaata ngaru.
Ki te hiahia koe, ka taea hoki e au te tāpiri i ngā tauira rapanga me te whakaoti i ngā kōrero (hei tauira, te whakatau i te arotahi, te āhua rerekē, te tuhi rānei i tētahi tātuhi o tētahi porowhita mai i tana whārite).
