Ngā Tauira Pātai e Matapaki ana i te Whārite o te Raina Pānga ki te Porowhita
Pendahuluan
He kaupapa nui te whārite o te pātata ki te porowhita i roto i te āhuahanga tātari. Mā te mārama ki te whakatau i te whārite o te pātata ki te porowhita ka āwhina i te whakaoti rapanga pāngarau maha i ngā taumata waenga ki te taumata matatau. Ka matapakihia e tēnei tuhinga ngā tauira rapanga me ngā tikanga mō te whakatau i te whārite o te pātata ki te porowhita.
Whakamāramatanga me te Ariā Taketake
Ko te tikanga, ka taea te whakaatu i tētahi porowhita i roto i te papa taunga mā te whakamahi i tētahi whārite tapawhā:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
ko \((a, b)\) te pūwāhi pokapū o te porowhita, ā, ko \(r\) te pūtoro o te porowhita.
Ko te pātata ki tētahi porowhita mai i tētahi pūwāhi o waho he rārangi e pā ana ki te porowhita i te pūwāhi kotahi tonu. Mēnā ka whakaarohia ko te whārite kei te rārangi:
\[ y = mx + c \]
kātahi ka taea te whakaatu i te āhua e whai ake nei ko te rārangi \(y = mx + c\) ka pātata ki te porowhita:
\[ \sqrt{(a + bm)^2 – r^2} = |c| \]
Ko \(m\) te pikinga o te rārangi pātapa, ā, ko \(c\) he pūmau.
Tātai Whārite Raina Pānga
Mō te pātata ki tētahi porowhita me te pokapū \((0,0)\) me te radius \(r\), ko tōna whārite i te pūwāhi \((x_1, y_1)\) i runga i te porowhita ko:
\[ x_1x + y_1y = r^2 \]
Engari, ki te mea kei te pūwāhi \((a, b)\ te pokapū o te porowhita, ko te whārite tēnei:
\[ (x_1 – a)(x – a) + (y_1 – b)(y – b) = r^2 \]
Ngā Tauira Pātai me ngā Kōrerorero
Pātai 1
He porowhita kei waenganui i te \((3, 4)\) me te radius e 5 waeine. Whakatauhia te whārite o te rārangi pātapa mai i te pūwāhi \((6, 8)\).
Kōrero:
I tēnei raruraru, ka hoatu he porowhita ki a tātou me te pokapū o \((3, 4)\), he radius o te 5, ā, me kimi e tātou te whārite o te rārangi pātapa mai i te pūwāhi \((6, 8)\). Anei ngā mahi hei whakaoti:
1. Manatokohia kāore te pūwāhi \((6, 8)\) i roto i te porowhita: \\
\[
\sqrt{(6-3)^2 + (8-4)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5
\]
Nō reira, kei runga i te porowhita te pūwāhi, nō reira ka taea te whakamahi hei kimi i te rārangi pātapa.
2. Mā te whakamahi i te tātai:
\[
(x_1 – a)(x – a) + (y_1 – b)(y – b) = r^2
\]
\[
(6 – 3)(x – 3) + (8 – 4)(y – 4) = 5^2
\]
3. Whakangāwaritia te whārite:
\[
3(x – 3) + 4(y – 4) = 25
\]
4. Whakawhanake:
\[
3x – 9 + 4y – 16 = 25
\]
\[
3x + 4y – 25 = 50
\]
Ko te whārite o te rārangi pātata i whiwhihia ko:
\[
3x + 4y = 50
\]
Pātai 2
Tātaitia te whārite \[x^2 + y^2 = 16\]. Whakatauhia te whārite o te rārangi pātapa mai i te pūwāhi \((4, 0)\).
Kōrero:
He porowhita kei waenganui i te \((0, 0)\) me te radius e 4 waeine. Ko te pūwāhi o waho kua hoatu ko \((4, 0)\).
1. Mā te whakamahi i te tātai mō tētahi porowhita me te pokapū \((0, 0)\):
\[
x_1x + y_1y = r^2
\]
2. Whakakapinga uara:
\[
4x + 0 \cdot y = 4^2
\]
3. Whakangāwaritia:
\[
4x = 16
\]
\[
x = 4
\]
Ko te tikanga, ko te whārite pānihi kei a tātou ko:
\[
x = 4
\]
Pātai 3
Hangaia he rārangi e pātata ana ki te porowhita \((x+2)^2 + (y-3)^2 = 9\) i te pūwāhi \((-1, 5)\).
Kōrero:
He porowhita kei waenganui i te \((-2, 3)\) me te pūtoro e 3 waeine. Ko te pūwāhi o te pātakitanga ko \((-1, 5)\).
1. Manatokohia kei runga i te porowhita te pūwāhi:
\[
((-1+2)^2 + (5-3)^2) = 1 + 4 = 5 \neq 9 \rightarrow \text{kāore i runga i te porowhita}
\]
Nō reira, tera pea he hapa kei roto i te whakaurunga o te pātai, o te tohu rānei. Mena ka hoatuhia te \((-2, 6)\) hei tohu o te pātakitanga:
2. Whakangāwaritia:
\[
(x_1-a)(xa) + (y_1-b)(yb) = r^2
\]
Whakakapinga:
\[
(-2-(-2))(x+2) + (6-3)(y-3) = 9\]
Hua:
\[
0 + 3(y-3) = 3^2
\]
\[
3y-9=9
\]
\[
y = 6
Ko te hua o tēnei ko te kōrero me ngā kōrero:
He tino tika a y = 6.
Huinga tirotiro whakatau, maumahara, e rua ngā huarahi tatau.
tika/hē..Tērā Tāpiri whakarāpopototanga gar. td 2; manatokohia te āhua me te (whatunga). tāpirihia te taha katoa o te kapinga.
Koinā katoa, ngā mihi nui ki a koutou, ā, ko te tumanako ka whai hua tēnei mā ōku hoa.
Hei tāpiritanga, akohia te ariā o te whakapakari i te māramatanga.
Ko te hunga kāore i mārama i mua, ki te pai te Atua, ka whai i ngā mahi pokanga taipitopito ake.
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