Ajụjụ Ihe Nlereanya Na-atụle Gburugburu na Tangents
Okirikiri bụ isiokwu dị mkpa na geometry matrix, ebe a na-egosipụta echiche miri emi gbasara anya, akụkụ, na ọdịdị. Otu echiche a na-ekwukarị n'isiokwu a bụ ahịrị tangent na okirikiri. N'isiokwu a, anyị ga-atụle ọtụtụ nsogbu ihe atụ metụtara okirikiri na tangent.
Nghọta Isi nke Gburugburu na Tangents
Lingkaran
Okirikiri bụ ọdịdị geometric nke e ji ihe niile dị n'ime ụgbọelu kee, nke dị anya site na ebe a na-akpọ etiti okirikiri ahụ. A na-akpọ anya a kapịrị ọnụ nke okirikiri ahụ.
Tangent
Tangent na okirikiri bụ ahịrị nke metụrụ okirikiri ahụ kpọmkwem n'otu ebe. A na-akpọ ebe a isi ihe nke tangency. Tangent nwere ọtụtụ ihe dị mkpa, gụnyere:
– Ahịrị tangent ahụ na-adị mgbe niile n'akụkụ okirikiri ahụ n'ebe tangency dị.
– Ogologo tangent site na isi ihe dị n'èzí okirikiri ahụ ruo na okirikiri ahụ bụ otu ihe ahụ ma ọ bụrụ na e sere tangent abụọ site na isi ihe ahụ.
Ajụjụ na Mkparịta ụka Ndị Dị Mkpa
N'okpuru ebe a, anyị ga-enye ọtụtụ ajụjụ atụ ndị na-akọwa echiche nke okirikiri na tangent n'ụzọ zuru ezu.
Ihe atụ Ajụjụ nke 1: Ịchọta Ogologo Ahịrị Tangent
Ajụjụ:
E nyere okirikiri nwere etiti \(O\) na radius \(r = 6 \, \text{cm}\). Site na isi \(P\) n'èzí okirikiri ahụ nke dị cm 10 site na etiti okirikiri ahụ, a na-adọta tangent abụọ \(PA\) na \(PB\) na okirikiri ahụ. Gbakọọ ogologo tangent \(PA\).
Azịza:
N'okwu a, anyị nwere ike iji usoro Pythagorean. See triangle \(\triangle OAP\):
– \(OP = 10 \, \text{cm}\) (anya site na isi mpụta ruo etiti okirikiri ahụ)
– \(OA = 6 \, \text{cm}\) (radius nke okirikiri ahụ)
– \(PA\) bụ ahịrị tangent nke a ga-achọtarịrị
\[
OP^2 = OA^2 + PA^2
\]
\[
10^2 = 6^2 + PA^2
\]
\[
100 = 36 + PA^2
\]
\[
PA^2 = 64
\]
\[
PA = \sqrt{64} = 8 \, \ederede{cm}
\]
Ya mere, ogologo nke ahịrị tangent \(PA\) bụ 8 cm.
Ihe atụ Ajụjụ nke 2: Ịchọta Isi Ihe Na-agbanwe Agbanwe
Ajụjụ:
E nyere okirikiri nwere nha anya \((x – 3)^2 + (y – 4)^2 = 25\) na ahịrị \(y = 2x + 1\). Chọpụta ebe tangency dị n'etiti okirikiri na ahịrị ahụ.
Azịza:
Nke mbụ, anyị na-achọpụta etiti na radius nke okirikiri ahụ:
– Ebe etiti \(O(3, 4)\)
– Radius \(r = \sqrt{25} = 5\)
Iji chọta isi ihe nke tangency, ka anyị were ya na isi ihe nke tangency bụ \(T(x_1, y_1)\) nke dịkwa n'ahịrị \(y = 2x + 1\). Mgbe ahụ:
\[
y_1 = 2x_1 + 1
\]
\(T(x_1, y_1)\) ga-emezukwa nha anya nke okirikiri ahụ:
\[
(x_1 – 3)^2 + (y_1 – 4)^2 = 25
\]
Tinye \(y_1 = 2x_1 + 1\) n'ime nha okirikiri:
\[
(x_1 – 3)^2 + ((2x_1 + 1) – 4)^2 = 25
\]
\[
(x_1 – 3)^2 + (2x_1 – 3)^2 = 25
\]
Anyị kwesịrị ịgbakọ akụkụ abụọ.
\[
(x_1 – 3)^2 = x_1^2 – 6x_1 + 9
\]
\[
(2x_1 – 3)^2 = 4x_1^2 – 12x_1 + 9
\]
Jikọta nsonaazụ abụọ a:
\[
x_1^2 – 6x_1 + 9 + 4x_1^2 – 12x_1 + 9 = 25
\]
\[
5x_1^2 – 18x_1 + 18 = 25
\]
Wepụ 25 site n'akụkụ abụọ ahụ:
\[
5x_1^2 – 18x_1 – 7 = 0
\]
Dozie usoro nhazi quadratic:
\[
x_1 = \frac{18 \pm \sqrt{18^2 + ugboro anọ 5 \u003d 7}}{2 \u003d 5}
\]
\[
x_1 = \frac{18 \pm \sqrt{324 + 140}}{10}
\]
\[
x_1 = \frac{18 \pm \sqrt{464}}{10}
\]
\[
x_1 = \frac{18 \pm 2\sqrt{116}}{10}
\]
\[
x_1 = \frac{18 \pm 2\sqrt{4 \ugboro 29}}{10}
\]
\[
x_1 = \frac{18 \pm 4\sqrt{29}}{10}
\]
\[
x_1 = 1.8 \pm 0.4 \ sqrt{29}
\]
Gbakọọ uru nke \(y_1\):
Nke na-emeju y = 2x + 1:
– Ọ bụrụ na \(x_1 = 1.8 + 0.4\sqrt{29}\), mgbe ahụ \(y_1 = 2(1.8 + 0.4\sqrt{29}) + 1\)
– Ọ bụrụ na \(x_1 = 1.8 – 0.4\sqrt{29}\), mgbe ahụ \(y_1 = 2(1.8 – 0.4\sqrt{29}) + 1\)
Nyocha:
Ya mere, anyị na-enweta isi ihe abụọ nke njikọ nke nha anya nke okirikiri ahụ na ahịrị ahụ.
Ajụjụ Ihe Nlereanya nke 3: Ịchọpụta Nha nhata nke Ahịrị Tangent
Ajụjụ:
E nyere okirikiri nwere nha nha \((x – 2)^2 + (y – 3)^2 = 20\). Chọpụta nha nha ahịrị tangent na okirikiri nke gafere isi \(6, 7)\).
Azịza:
Enwere ike ịchọta tangent na okirikiri nwere etiti \((h, k)\) na radius \(r\) site na isi ihe mpụta a maara site na nha nhata:
Ahịrị tangent ahụ na-agafe ebe mpụta \((x_1, y_1)\):
\[
(x - 2) (x_1 - 2) + (y - 3) (y_1 - 3) = 20
\]
Dochie isi ihe dị n'èzí \(6, 7)\):
\[
(x – 2) (6 – 2) + (y – 3) (7 – 3) = 20
\]
\[
4(x – 2) + 4(y – 3) = 20
\]
\[
4(x – 2 + y – 3) = 20
\]
\[
4x + 2y -20 = 20
\]
\[
4x + 4y -20 = 20
\]
\[
x + y = 5
\]
Usoro nke ahịrị tangent bụ:
\[
x + y = 9
\]
Ya mere, mgbanwe nke nha nhata nke ahịrị ahụ site na isi nke ahịrị okirikiri buru ibu nke ukwuu ma nwee ike ịgbanwe dabere na nsonaazụ ma ọ bụ ihe ngosi anya.
Mmechi
Mkparịta ụka nke okirikiri na tangents na-ekpuchi ọtụtụ akụkụ dị mkpa nke mgbakọ na mwepụ, site na itinye usoro ndị bụ isi dịka Pythagorean theorem ruo na idozi nha nha quadratic. Site na ihe atụ ndị a, anyị nwere ike ịghọta nke ọma otu esi etinye echiche ndị a n'ọrụ n'ọnọdụ dịtụ mgbagwoju anya. Enwere m olileanya na isiokwu a enyerela aka inye nkọwa doro anya nke otu esi emeso ma dozie nsogbu ndị metụtara okirikiri na tangent.
