Tauira o tētahi Pātai Kōrero mō te Whārite o tētahi Porowhita
He kaupapa nui te whārite o te porowhita i roto i te āhuahanga tātari. He tino whai hua te māramatanga pai ki te whārite o te porowhita, ehara i te mea i roto i te pāngarau anake engari i roto hoki i ngā tono hangarau me te pūtaiao. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira o ngā whārite o te porowhita me ō rātou otinga. Ko te whāinga he whakarato i tētahi tirohanga whānui me te mārama mō te whakaoti rapanga e pā ana ki ngā whārite o te porowhita.
Whārite Whānui o te Porowhita
Ko te whārite tino noa o tētahi porowhita i roto i ngā taunga Cartesian ko:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
Kei hea:
– Ko \( (a, b) \) ngā taunga o te pokapū o te porowhita.
– Ko te \( r \) te pūtoro o te porowhita.
Mena kei te pūwāhi \( (0, 0) \ te pokapū o te porowhita), ka pēnei te whārite o te porowhita:
\[ x^2 + y^2 = r^2 \]
Nā, me matapaki tātou i ētahi tauira pātai me ā rātou otinga.
Tauira Pātai 1
Pātai: Whakatauhia te whārite o tētahi porowhita ko tōna pokapū kei te pūwāhi (3, -2) ā, ko tōna radius he 5.
Otinga:
Whakamahia te tātai whānui mō te whārite o tētahi porowhita:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
Whakakapia ngā uara \( a = 3 \), \( b = -2 \), me \( r = 5 \):
\[ (x – 3)^2 + (y + 2)^2 = 5^2 \]
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Nā, ko te whārite o te porowhita koia tēnei:
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Tauira Pātai 2
Pātai: Whakatauhia te whārite o tētahi porowhita ko tōna pokapū kei te pūtake (0, 0) ā, ko tōna radius he 7.
Otinga:
Nā te mea kei te pūtake te pokapū o te porowhita, ka taea e tātou te whakamahi i te whārite māmā:
\[ x^2 + y^2 = r^2 \]
Whakakapia te uara \( r = 7 \):
\[ x^2 + y^2 = 7^2 \]
\[ x^2 + y^2 = 49 \]
Nā, ko te whārite o te porowhita koia tēnei:
\[ x^2 + y^2 = 49 \]
Tauira Pātai 3
Pātai: Whakatauhia te whārite o tētahi porowhita ko tōna pokapū kei te pūwāhi (4, -5) ā, e pā ana ki te tuaka Y.
Otinga:
Ko te tikanga o te porowhita e pātata ana ki te tuaka-Y ko te tawhiti mai i te pokapū o te porowhita ki te tuaka-Y he rite ki tōna pūtoro. Ko tēnei tawhiti te uara tino o te taunga-X o te pokapū o te porowhita. Nō reira, ko te pūtoro he 4.
Whakamahia te tātai whānui mō te whārite o tētahi porowhita:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
Whakakapia ngā uara \( a = 4 \), \( b = -5 \), me \( r = 4 \):
\[ (x – 4)^2 + (y + 5)^2 = 4^2 \]
\[ (x – 4)^2 + (y + 5)^2 = 16 \]
Nā, ko te whārite o te porowhita koia tēnei:
\[ (x – 4)^2 + (y + 5)^2 = 16 \]
Tauira Pātai 4
Pātai: Kei te porowhita te whārite \( x^2 + y^2 – 6x + 4y – 12 = 0 \). Tātaihia te pokapū me te pūtoro o te porowhita.
Otinga:
Hei whakaoti i tēnei whārite, me huri ki te āhua paerewa \( (x – a)^2 + (y – b)^2 = r^2 \). Ko ngā mahi hei whakaoti koia ēnei:
1. Te whakarōpū me te whakaoti rapanga tapawhā tino tika:
Ko te whārite tuatahi ko:
\[ x^2 + y^2 – 6x + 4y – 12 = 0 \]
Rōpū \( x \) me \( y \):
\[ (x^2 – 6x) + (y^2 + 4y) = 12 \]
2. Whakaotia te tapawhā tino tika:
Mō \( x^2 – 6x \):
\[ x^2 – 6x + 9 \]
Mō \( y^2 + 4y \):
\[ y^2 + 4y + 4 \]
Tāpirihia te 9 me te 4 ki ngā taha e rua o te whārite:
\[ (x^2 – 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4 \]
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Nā, ko te whārite o tētahi porowhita i te āhua paerewa koia tēnei:
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Mai i konei, ka kitea ko te pokapū o te porowhita ko \( (3, -2) \) ā, ko te radius ko \( r = \sqrt{25} = 5 \).
Tauira Pātai 5
Pātai: Tāutuhia te whārite o te porowhita e tika ana mā roto i ngā pūwāhi (2, 3) me (4, 5), ā, ko tōna pokapū kei te rārangi x = 3.
Otinga:
Mai i te pātai, e mōhio ana tātou ko te pokapū o te porowhita ko (3, b). Ka haere hoki te porowhita mā roto i ngā pūwāhi e rua e mōhiotia ana. Nā te mea ka haere te porowhita mā roto i (2, 3), ko te tawhiti mai i te pokapū ki tēnei pūwāhi ko te pūtoro.
Ko te whārite o tētahi porowhita koia tēnei:
\[ (x – 3)^2 + (y – b)^2 = r^2 \]
Pūwāhi whakakapinga (2, 3):
\[ (2 – 3)^2 + (3 – b)^2 = r^2 \]
\[ 1 + (3 – b)^2 = r^2 \]
\[ (3 – b)^2 = r^2 – 1 \]
Pūwāhi whakakapinga (4, 5):
\[ (4 – 3)^2 + (5 – b)^2 = r^2 \]
\[ 1 + (5 – b)^2 = r^2 \]
\[ (5 – b)^2 = r^2 – 1 \]
Mai i ngā whārite e rua, e mōhio ana tātou (3 – b)^2 = (5 – b)^2. Nō reira:
\[ 3 – b = \pm(5 – b) \]
Mena ko \( 3 – b = 5 – b \), kāore e taea te pono o te hua. Nō reira:
\[ 3 – b = -(5 – b) \]
\[ b = 4 \]
Me te b = 4, ko te whārite o te porowhita koia tēnei:
\[ (x – 3)^2 + (y – 4)^2 = 2 \]
Heoi, ka taea e tātou te tatau i te pūtoro r mai i te tawhiti i waenganui i te pokapū me te pūwāhi (2, 3) = \(\sqrt{(2 – 3)^2 + (3 – 4)^2} \) = \(\sqrt{1+1}\) = \(\sqrt {2}\)
Ko te whārite o te porowhita ko:
\[ (x – 3)^2 + (y – 4)^2 = 2 \]
Whakamutunga
Mā te mārama ki te whārite o te porowhita ka māmā ake te whakaoti rapanga pāngarau maha. I ia take, he mea nui te tautuhi i te pokapū me te radius. Ko te tumanako, mā ēnei tauira rapanga me ō rātou whakamārama ka whakamārama, ka āwhina hoki i a koe ki te ako i te whārite o te porowhita. Mā te mahi ka tino pai te pāngarau, nō reira kaua e mangere ki te whakamātau i ngā momo rapanga hei whakapai ake i ō pūkenga.
