Ngā Tauira Pātai me te Kōrero mō ngā Porowhita me ngā Pānga
Ko ngā porowhita me ngā tātai he kaupapa e rua e kōrerohia pinepinetia ana i roto i te pāngarau, inā koa i te taumata kura tuarua. He mea nui te mārama ki te ariā me te whakamahinga o ngā tātai ki ngā porowhita hei whakahōhonu ake i tō mōhiotanga ki te āhuahanga. Ka whakaratohia e tēnei tuhinga he tauira rapanga me ngā kōrero mō ngā porowhita me ngā tātai hei whakarato i te māramatanga hōhonu ake ki ngā kaipānui.
He Kupu Whakataki ki te Ariā o ngā Porowhita me ngā Pānga
Porowhita
Ko te porowhita he huinga pūwāhi i roto i te papa e rite ana te tawhiti mai i tētahi pūwāhi pumau e kiia nei ko te pokapū o te porowhita. Ko tēnei tawhiti pumau e mōhiotia ana ko te pūtoro o te porowhita. I roto i te pāngarau, ka taea te tautuhi i tētahi porowhita mā te whārite:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
ko \((a, b)\) ngā taunga o te pokapū o te porowhita, ā, ko \(r\) te pūtoro.
Pānga
Ko te pātata ki te porowhita he rārangi e pā ana ki te porowhita i te pūwāhi kotahi. Ka kiia tēnei pūwāhi ko te pūwāhi pātata. Ko te āhuatanga matua o te pātata ko tōna tū poutū ki te pūtoro i tuhia mai i te pokapū o te porowhita ki te pūwāhi pātata.
Ngā Pātai Tauira me te Kōrero
Pātai 1: Te Whakatau i te Whārite o te Raina Pānga
Pātai:
Hoatu he porowhita me te pokapū kei \( (2, 3) \) me te radius 5. Whakatauhia te whārite o te rārangi pātapa ki te porowhita i te pūwāhi \( P \) me ngā taunga \( (5, 7) \).
Kōrero:
Hipanga 1: Kia tino mohio kei runga te pūwāhi \(P \) i te porowhita.
Hei tirotiro mēnā kei runga a \( P (5, 7) \) i tētahi porowhita me te pokapū \( (2, 3) \) me te radius \( 5 \), tāpirihia ngā taunga o \( P \) ki roto i te whārite o te porowhita:
\[ (x – 2)^2 + (y – 3)^2 = 5^2 \]
\[ (5 – 2)^2 + (7 – 3)^2 = 25 \]
\[ 3^2 + 4^2 = 25 \]
\[ 9 + 16 = 25 \]
Nā te mea he pono te ōritetanga, kei runga tonu i te porowhita te pūwāhi \( P \).
Hipanga 2: Whakatauhia te pikinga o te pūtoro e haere ana i roto i \( (2, 3) \) me \( (5, 7) \):
\[ m_{radius} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{7 – 3}{5 – 2} = \frac{4}{3} \]
Hipanga 3: Ko te rōnaki o te rārangi pātapa e poutū ana ki te rōnaki o te pūtoro (ko te rōnaki o te hua he -1):
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]
Hipanga 4: Whakatauhia te whārite o te rārangi pātata mā te whakamahi i te pūwāhi \( P (5, 7) \):
\[ y – y_1 = m (x – x_1) \]
\[ y – 7 = -\frac{3}{4} (x – 5) \]
Whakangāwaritia:
\[ y – 7 = -\frac{3}{4}x + \frac{15}{4} \]
\[ 4y – 28 = -3x + 15 \]
\[ 3x + 4y – 43 = 0 \]
Nā, ko te whārite o te rārangi pātata ko:
\[ 3x + 4y – 43 = 0 \]
Pātai 2: Te Whakatau i te Pūwāhi o te Tātaitanga mai i te Whārite Raina
Pātai:
Hoatu he porowhita me te whārite \( x^2 + y^2 = 25 \) me te rārangi \( y = \frac{3}{4}x + 2 \). Tātaihia te pūwāhi pātata i waenganui i te rārangi me te porowhita.
Kōrero:
Hipanga 1: Whakakapia te whārite o te rārangi ki te whārite o te porowhita:
Whārite o te porowhita:
\[ x^2 + y^2 = 25 \]
Whakakapia \( y = \frac{3}{4}x + 2 \) ki roto i te whārite porowhita:
\[ x^2 + \left(\frac{3}{4}x + 2\right)^2 = 25 \]
\[ x^2 + \left(\frac{9}{16}x^2 + \frac{12}{4}x + 4 \right) = 25 \]
\[ x^2 + \frac{9}{16}x^2 + \frac{6}{2}x + 4 = 25 \]
\[ x^2 + \frac{9}{16}x^2 + 3x + 4 = 25 \]
Hipanga 2: Whakangāwaritia te whārite:
\[ 16x^2 + 9x^2 + 48x + 64 = 400 \]
\[ 25x^2 + 48x + 64 – 400 = 0 \]
\[ 25x^2 + 48x – 336 = 0 \]
Hipanga 3: Te kimi i ngā pūtake mā te whakamahi i te tātai tapawhā:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
\[ a = 25, b = 48, c = -336 \]
\[ x = \frac{-48 \pm \sqrt{48^2 – 4 \cdot 25 \cdot (-336)}}{2 \cdot 25} \]
\[ x = \frac{-48 \pm \sqrt{2304 + 33600}}{50} \]
\[ x = \frac{-48 \pm \sqrt{35904}}{50} \]
\[ x = \frac{-48 \pm 189.501}{50} \]
Te tīpako i tētahi \( x \) tika i runga i te pūwāhi pātata (kotahi anake \( x \) ka puta he pūwāhi pātata):
\[ x = \frac{141.501}{50} \tata ki te 2.83 \]
\[ x \tata ki te 2.83 \]
Hipanga 4: Tāpirihia te \( x \) ki te whārite o te rārangi hei whiwhi i te \( y \):
\[ y = \frac{3}{4}(2.83) + 2 \]
\[ y \tata ki te 2.12 + 2 \]
\[ y \ tata ki te 4.12 \]
Nō reira, ko te pūwāhi pātata i waenganui i te rārangi \( y = \frac{3}{4}x + 2 \) me te porowhita \( x^2 + y^2 = 25 \) ko \( (2.83, 4.12) \).
Whakamutunga
Ko te matatau ki ngā ariā o ngā porowhita me ngā pātapa e uru ana ki te mārama ki ngā kaupapa matua o te āhuahanga me te kaha ki te whakaoti rapanga mā te whakamahi i ngā whārite pāngarau. Ka āwhina ngā rapanga pēnei i ngā mea o runga ake nei i ngā ākonga ki te whakamahi i te ariā i roto i ngā āhuatanga tuturu. Mā te mahi tonu, e tumanakohia ana ka mārama ake, ka whakaoti rapanga ngāwari ake ngā ākonga.
