Whārite o te Raina Pānga ki te Porowhita
Ko te porowhita tētahi o ngā mea āhuahanga tino taketake, ā, e kitea pinepinetia ana i roto i ngā momo mara pūtaiao, mai i te pāngarau taketake ki te miihini ā-iwi me te hoahoanga. Ko tētahi o ngā ariā matua e pā ana ki ngā porowhita i roto i te āhuahanga tātari ko te whārite o te rārangi pātata ki te porowhita. Mā te mārama ki te whārite o te rārangi pātata ki te porowhita ka tuwhera ake te māramatanga hohonu ake ki ngā whanaungatanga i waenga i ngā mea āhuahanga me ō rātou whakamahinga i roto i te oranga o ia rā. Ka whakamāramahia e tēnei tuhinga te whārite o te rārangi pātata ki te porowhita me te taipitopito, mai i te ariā taketake me te tango i te whārite, me te whakamahi i ngā tauira.
Te Ariā Taketake o te Pānga ki te Porowhita
Ko te pātata ki te porowhita he rārangi e pā ana ki te porowhita i te pūwāhi kotahi anake, engari kāore e whakawhiti. Ko te pūwāhi e tūtaki ai te rārangi me te porowhita ka kiia ko te pūwāhi pātata. He rerekē ki ngā rārangi e whakawhiti noa ana i te porowhita i ngā pūwāhi e rua, ko te āhuatanga ahurei o ngā pātata ko ia pātata ki te porowhita he poutū ki te pūtoro o te porowhita i taua pūwāhi.
Ngā Whārite Whānui o ngā Porowhita me ngā Rārangi
I mua i te matapaki i te whārite o te rārangi pātapa, me mōhio tuatahi ki te whārite whānui o te porowhita me te rārangi i roto i ngā taunga Cartesian.
Whārite Porowhita
Ko te whārite o tētahi porowhita kei te pūwāhi \((h, k)\) te pokapū me te radius \(r\):
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Whārite Raina
Ka taea te whakaatu i ngā rārangi i te papa Cartesian i roto i ngā āhua maha, ko tētahi o ngā mea tino kitea ko te āhua o te pikinga-taunga:
\[ y = mx + c \]
ko \(m\) te pikinga (te pikinga rānei) o te rārangi, ā, ko \(c\) te haukoti (te tapahi) huri noa i te tuaka-y.
Te Whakatau i te Whārite o te Raina Pātene ki te Porowhita
He maha ngā tikanga ka taea te whakamahi hei whakatau i te whārite o te rārangi pātapa ki te porowhita. Anei ētahi o ngā tikanga tino noa.
Tikanga 1: Te Whakamahi i ngā Pūwāhi Rōrahi me ngā Pūwāhi Pānga
Mena e mōhio ana tātou ki te pūwāhi pātata \((x_1, y_1)\) i runga i tētahi porowhita kei waenganui \((h, k)\), ka taea e tātou te whakamahi i te āhuatanga āhuahanga e whakaatu ana ko te rārangi pātata he poutū ki te pūtoro o te porowhita i te pūwāhi pātata. Mena ko te rōnaki o te pūtoro e haere ana i roto i ngā pūwāhi \((h, k)\) me \((x_1, y_1)\) ko:
\[ m_{radius} = \frac{y_1 – k}{x_1 – h} \]
Kātahi ko te rōnaki o te rārangi pātapa, e poutū ana ki te rārangi pūtoro, ko:
\[ m_{tāngātahi} = -\frac{1}{m_{rādius}} = -\frac{x_1 – h}{y_1 – k} \]
Kia mōhiotia te pikinga o te rārangi pātata, ka taea e tātou te tuhi i te whārite o te rārangi pātata ki te āhua o te pikinga-taupoki mā te whakamahi i te pūwāhi \((x_1, y_1)\):
\[ y – y_1 = m_{tāngā}(x – x_1) \]
I roto rānei i te puka paerewa:
\[ y – y_1 = -\frac{x_1 – h}{y_1 – k}(x – x_1) \]
Tikanga 2: Te Whakamahi i te Whakakapinga me te Whakawehewehe
Hei kimi i tētahi pātata ki tētahi porowhita e mōhiotia ana mā te whakamahi i te tikanga whakakapinga me te tikanga wehewehe, ka tīmata mā te tuhi i te whārite o te porowhita me te whakauru i te whārite whānui o te rārangi. Ko te whārite whānui o tētahi rārangi ko \( y = mx + c \). Mā te whakakotahi i tēnei ki te whārite o te porowhita:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Whakakapia te \( y \) i roto i te whārite porowhita ki te \( mx + c \):
\[ (x – h)^2 + (mx + c – k)^2 = r^2 \]
Kātahi ka whakawhānuihia tēnei whārite ki te āhua tapawhā paerewa \(Ax^2 + Bx + C = 0\). Kia pātata ai te rārangi ki te porowhita, me kotahi tonu te otinga mō \(x\), nō reira me ōrite te wehewehe o te whārite tapawhā ki te kore. Ko te wehewehe o te whārite tapawhā \(Ax^2 + Bx + C = 0\) ko:
\[ D = B^2 – 4AC \]
Mā te \(D = 0\), ka taea e tātou te whakatau i ngā uara o \(m\) me \(c\) e hono ana te rārangi ki te porowhita.
Ngā Tauira Taupānga
Tauira 1: Te Whakatau i te Whārite o te Raina Pānga
Me kī he porowhita tā tātou me te whārite \( (x – 3)^2 + (y + 4)^2 = 25 \) ā, e hiahia ana tātou ki te mōhio ki te whārite o te rārangi pātapa e tika ana mā te pūwāhi \((-1, 5)\).
Tuatahi, ka tirohia mēnā kei runga i te porowhita te pūwāhi. Mā te tāpiri i te \((x, y) = (-1, 5)\) ki te whārite o te porowhita:
\[ (-1 – 3)^2 + (5 + 4)^2 = (-4)^2 + 9^2 = 16 + 81 = 97 \]
Nā te mea kāore tēnei pūwāhi i te porowhita. Heoi anō, ka kitea tonutia he rārangi e tika ana mā tēnei pūwāhi, ā, e poutū ana ki te pūtoro i te pūwāhi o te pātata.
Tuatahi, ka kitea e tātou te pikinga o te pūtoro e tika ana mā te pūwāhi:
\[ m_{radius} = \frac{5 + 4}{-1 – 3} =\frac{9}{-4} = -\frac{9}{4} \]
Nō reira, ko te pikinga o te rārangi pātapa ko:
\[ m_{tangent} = -\frac{1}{m_{radius}} = \frac{4}{9} \]
Ko te whārite o te rārangi pātata e whakamahi ana i tēnei rōnaki me te haere mā te pūwāhi \((-1,5)\) ko:
\[ y – 5 = \frac{4}{9}(x + 1) \]
Whakamutunga
He ariā āhuahanga tino taketake te whārite o te pātata ki te porowhita, engari he whānuitia ngā whakamahinga i roto i ngā momo mara. Mā te mārama ki ngā āhuatanga o ngā pātata me ngā tikanga mō te whakatau i ā rātou whārite, ka taea e tātou te whakamahi i tēnei ariā hei whakaoti rapanga maha i roto i te pāngarau me te pūtaiao.
Mā te mārama ki ngā porowhita me ngā tātaitanga ka whānui ake ngā māramatanga ki te whanaketanga o te pūtaiao, inā koa i roto i te pāngarau tātari. Mā te huarahi pūnaha, ka taea e tātou te hono i ngā momo huānga i roto i te wāhi rua-ahu, ka whakapakari ake i tō tātou māramatanga ki ngā kaupapa matua o te āhuahanga ka taea te whakamahi hei tūāpapa mō te tūhuratanga atu i roto i te āhuahanga me te tātari ā-wāhi.
