Ngā Tauira Pātai e Matapaki ana i ngā Wāhanga Kōnika Porowhita
Pendahuluan
He pūtaiao taketake te pāngarau e whai wāhi nui ana ki ngā āhuatanga maha o te oranga tangata. Ko tētahi kaupapa tino wero i roto i te pāngarau ko te āhuahanga, inā koa ko ngā wāhanga kōniko. I roto i tēnei tuhinga, ka matapakihia e mātou tētahi o aua wāhanga kōniko: te porowhita. Ka whakaratohia e tēnei tuhinga he tauira rapanga me tētahi matapakinga whānui mō ngā porowhita, e tumanakohia ana ka āwhina i ngā ākonga ki te mārama ake ki tēnei kaupapa.
Te Whakamāramatanga me ngā Āhuatanga o ngā Porowhita
I mua i tā tātou urunga atu ki ngā tauira pātai, he mea āwhina kia mārama tuatahi tātou he aha te porowhita. Ko te porowhita te kohinga o ngā pūwāhi katoa i roto i te papa e pumau ana te tapeke o ngā tawhiti mai i ngā pūwāhi pumau e rua (ōna arotahi). Ka kiia ēnei pūwāhi pumau e rua ko ngā arotahi o te porowhita (F1 me F2).
I roto i te āhua taurangi, ka taea te whakaahua i tētahi porowhita mā tōna whārite whānui:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
ko \( a \) te tawhiti mai i te pokapū o te porowhita ki te pūwāhi tawhiti rawa atu i te tuaka matua, ā, ko \( b \) te tawhiti mai i te pokapū o te porowhita ki te pūwāhi tawhiti rawa atu i te tuaka āwhina.
Ngā Tauira Pātai me te Kōrero mō ngā Porowhita
Pātai 1:
Ko te whārite o te porowhita ko \(\frac{x^2}{25} + \frac{y^2}{9} = 1\). Tātaihia te roa o te tuaka matua, te roa o te tuaka āwhina, me ngā taunga o ngā arotahi.
Kōrero:
Ko te whārite o te porowhita kua hoatu ko \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).
1. Whakatauhia te roa o te tuaka matua me te tuaka āwhina:
\[ a^2 = 25 \Rightarrow a = \sqrt{25} = 5 \]
\[ b^2 = 9 \Rightarrow b = \sqrt{9} = 3 \]
Nō reira, ko te roa o te tuaka matua \(= 2a = 2(5) = 10\).
Ko te roa o te tuaka āwhina \(= 2b = 2(3) = 6\).
2. Whakatauhia ngā taunga arotahi:
Kei te tuaka matua te arotahi o te porowhita i tawhiti atu i te pokapū o \(\sqrt{a^2 – b^2}\).
\[ c = \sqrt{a^2 – b^2} = \sqrt{25 – 9} = \sqrt{16} = 4 \]
Nā te mea ko te tuaka matua o tēnei porowhita te tuaka-x, ko ngā taunga arotahi ko:
\( (c, 0) \) me \( (-c, 0) \) rānei \( (4, 0) \) me \( (-4, 0) \).
Pātai 2:
Ki te hoatu he porowhita me te pokapū kei \( (0, 0) \) me te tuaka matua kei runga i te tuaka-x, he 12 te roa o te tuaka matua, ā, he 8 te roa o te tuaka āwhina. Whakatauhia te whārite o te porowhita.
Kōrero:
1. Mēnā ka roa te tuaka matua (2a = 12), ka:
\[ a = \frac{12}{2} = 6 \]
2. Mēnā ka roa te tuaka āwhina (2b = 8), ka:
\[ b = \frac{8}{2} = 4 \]
Ko te whārite o tētahi porowhita me te pokapū kei \( (0, 0) \) me te tuaka matua kei runga i te tuaka-x ko:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Tāpirihia a \( a \) me \( b \) ki te whārite:
\[ \frac{x^2}{6^2} + \frac{y^2}{4^2} = 1 \]
Nā, ko te whārite o te porowhita ko:
\[ \frac{x^2}{36} + \frac{y^2}{16} = 1 \]
Pātai 3:
Tātaihia te āhua rerekē o te porowhita \(\frac{x^2}{49} + \frac{y^2}{36} = 1\).
Kōrero:
Ko te āhua rerekē (\( e \)) o tētahi porowhita ka hoatuhia e te whārite:
\[ e = \frac{c}{a} \]
kei hea \( c = \sqrt{a^2 – b^2} \).
Mai i te whārite porowhita, ka whiwhi tātou:
\[ a^2 = 49 \Rightarrow a = 7 \]
\[ b^2 = 36 \Pere Matau b = 6 \]
Nā, ka kitea e tātou \( c \):
\[ c = \sqrt{a^2 – b^2} = \sqrt{49 – 36} = \sqrt{13} \]
Te āhua rerekē (\( e \)):
\[ e = \frac{c}{a} = \frac{\sqrt{13}}{7} \]
Nō reira, ko te āhua rerekē o te porowhita ko:
\[ e = \frac{\sqrt{13}}{7} \]
Pātai 4:
Mena kei te \( (-5, 0) \) me \( (5, 0) \) ngā pūwāhi arotahi e rua o te porowhita, ā, ko te roa o te tuaka matua o te porowhita he 12, whakatauhia te whārite o te porowhita.
Kōrero:
1. Whakatauhia \( a \) :
Ko te tuaka matua o Panmaßn g he 12, kātahi ka \( 2a = 12 \).
Nō reira \( a = \frac{12}{2} = 6 \).
2. Whakatauhia \( c \) :
Ko ngā pūwāhi arotahi e rua ko \( (-5, 0) \) me \( (5, 0) \), kātahi:
\[ c = 5 \]
3. Whakatauhia \( b \) :
Whakamahia te whanaungatanga \( c = \sqrt{a^2 – b^2} \):
\[ 5 = \sqrt{6^2 – b^2} \]
\[ 25 = 36 – b^2 \]
\[ b^2 = 36 – 25 \]
\[ b^2 = 11 \]
4. Whakahokia te whārite o te porowhita:
Ko te whārite o te porowhita ko:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Te whakakapi i te \( a \) me te \( b \):
\[ \frac{x^2}{6^2} + \frac{y^2}{\sqrt{11}^2} = 1 \]
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Nā, ko te whārite o te porowhita ko:
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Te Katinga
Mā te matapaki i ngā raruraru i runga ake nei, ka kitea e tātou ko te mārama ki ngā porowhita ehara i te ako noa i ā rātou whārite me ngā kauwhata, engari me pēhea hoki te hononga o ngā āhuatanga me ngā huānga o ngā porowhita ki a rātou anō. He tino whai hua te mōhio ki tēnei rauemi i roto i ngā momo mara tono, pērā i te ahupūngao, te whetū, me ētahi atu mara hangarau. Ko te tumanako, mā roto i ēnei tauira raruraru me ngā matapakinga, ka pai ake tō mārama ki ngā ariā me ngā tono taketake o ngā wāhanga porowhita.
I tuhia tēnei tuhinga i runga i te tumanako ka hōhonu ake te māramatanga mō ngā porowhita. Me mahi tonu, kaua e mangere ki te tūhura i ētahi atu raruraru e pā ana hei whakapai ake i ō pūkenga me ō mātauranga!
