Zitsanzo za Mafunso Okhudza Equation ya Mzere Wozungulira ndi Mzere
Pendauluan
Kuyerekeza kwa tangent ndi bwalo ndi nkhani yofunika kwambiri pakuwunika momwe zinthu zilili. Kumvetsetsa momwe mungadziwire equation ya tangent ndi bwalo kungathandize kuthetsa mavuto osiyanasiyana a masamu pamlingo wapakati mpaka wapamwamba. Nkhaniyi ikambirana za mavuto osiyanasiyana ndi njira zodziwira equation ya tangent ndi bwalo.
Tanthauzo ndi Chiphunzitso Choyambira
Bwalo mu ndege ya coordinate nthawi zambiri limatha kuyimiridwa ndi equation ya quadratic:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
kumene \((a, b)\) ndiye pakati pa bwalo ndipo \(r\) ndiye utali wa bwalo.
Mzere wozungulira kuchokera ku mfundo yakunja ndi mzere womwe umakhudza bwalo pamalo amodzi. Ngati tikuganiza kuti mzerewo uli ndi equation:
\[ y = mx + c \]
ndiye kuti mkhalidwe woti mzere \(y = mx + c\) ukhale wozungulira bwalo ukhoza kufotokozedwa motere:
\[ \sqrt{(a + bm)^2 – r^2} = |c| \]
Kumene \(m\) ndi gradient ya mzere wa tangent ndipo \(c\) ndi chosasintha.
Fomula Yoyesera Mzere wa Tangent
Pa mzere wozungulira womwe uli ndi pakati \((0,0)\) ndi radius \(r\), equation yake pamalo \((x_1, y_1)\) pa bwalo ndi:
\[ x_1x + y_1y = r^2 \]
Koma ngati pakati pa bwalo pali mfundo ya \((a, b)\), ndiye kuti equation ndi iyi:
\[ (x_1 – a)(x – a) + (y_1 – b)(y – b) = r^2 \]
Mafunso ndi Zokambirana za Zitsanzo
Funso 1
Bwalo lozungulira lokhala ndi pakati pa mayunitsi 5 (3, 4)) ndi ma radius. Dziwani equation ya mzere wa tangent kuchokera pa mfundo 6, 8).
Kukambirana:
Mu vutoli, tapatsidwa bwalo lokhala ndi pakati pa \((3, 4)\), radius ya 5, ndipo tiyenera kupeza equation ya mzere wa tangent kuchokera pa mfundo \((6, 8)\). Nazi njira zothetsera vutoli:
1. Tsimikizani kuti mfundo \((6, 8)\) siili mkati mwa bwalo: \\
\[
\sqrt{(6-3)^2 + (8-4)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5
\]
Kotero mfundo ili pa bwalo kuti ligwiritsidwe ntchito kupeza mzere wa tangent.
2. Kugwiritsa ntchito njira iyi:
\[
(x_1 – a)(x – a) + (y_1 – b)(y – b) = r^2
\]
\[
(6 – 3)(x – 3) + (8 – 4)(y – 4) = 5^2
\]
3. Fewetsani equation:
\[
3(x – 3) + 4(y – 4) = 25
\]
4. Pangani:
\[
3x – 9 + 4y – 16 = 25
\]
\[
3x + 4y – 25 = 50
\]
Equation ya mzere wa tangent womwe wapezeka ndi:
\[
3x + 4y = 50
\]
Funso 2
Zungulirani ndi equation \[x^2 + y^2 = 16\]. Dziwani equation ya mzere wa tangent kuchokera pa mfundo \((4, 0)\).
Kukambirana:
Bwalo lozungulira lokhala ndi pakati pa \((0, 0)\) ndi utali wa mayunitsi 4. Malo akunja operekedwa ndi \((4, 0)\).
1. Kugwiritsa ntchito njira ya bwalo lokhala ndi pakati \((0, 0)\):
\[
x_1x + y_1y = r^2
\]
2. Kusintha mtengo:
\[
4x + 0 \cdot y = 4^2
\]
3. Zosavuta:
\[
4x = 16
\]
\[
x = 4
\]
Izi zikutanthauza kuti, equation ya tangent yomwe tili nayo ndi iyi:
\[
x = 4
\]
Funso 3
Pangani mzere womwe uli pafupi ndi bwalo \((x+2)^2 + (y-3)^2 = 9\) pamalo \((-1, 5)\).
Kukambirana:
Bwalo lozungulira lokhala ndi pakati pa \((-2, 3)\) ndi ma radius 3 units. Mfundo ya tangency ndi \((-1, 5)\).
1. Tsimikizirani kuti mfundo ili pa bwalo:
\[
((-1+2)^2 + (5-3)^2) = 1 + 4 = 5 \neq 9 \rightarrow \text{not on circle}
\]
Chifukwa chake, pakhoza kukhala cholakwika pakulowetsa funso kapena mfundo. Ngati \((-2, 6)\) yaperekedwa ngati mfundo yolumikizirana:
2. Zosavuta:
\[
(x_1-a)(xa) + (y_1-b)(yb) = r^2
\]
Kulowa m'malo:
\[
(-2-(-2))(x+2) + (6-3)(y-3) = 9\]
Zotsatira:
\[
0 + 3(y-3) = 3^2
\]
\[
3y-9=9
\]
\[
y = 6 ndi
Zikuoneka kuti mfundo ndi maganizo ake ndi awa:
y = 6 ndi yolondola.
Mapeto onani kuphatikiza, kukumbukira, njira ziwiri zowerengera.
chovomerezeka/chabodza..Chifukwa chake kuphatikiza chidule cha gar. td 2; mawonekedwe otsimikizika ndi (gridi). plis perimeter all coverage btns.
Ndizo zonse, zikomo ndipo ndikukhulupirira kuti izi zithandiza anzanga.
Kuphatikiza apo, phunzirani mfundo yolimbikitsa kumvetsetsa.
Iwo omwe kale sanamvetse, Mulungu akalola, adzapatsidwa njira zambiri zofunika kwambiri zochitira opaleshoni.
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