Mibvunzo Yemuenzaniso Yekukurukura Madenderedzwa NemaTangents
Madenderedzwa inyaya inokosha mu matrix geometry, uko pfungwa dzakadzama pamusoro pedaro, makona, uye maumbirwo dzinoratidzwa. Imwe pfungwa inowanzo kurukurwa munyaya iyi ndeye mutsetse we tangent kuenda kudenderedzwa. Munyaya ino, tichakurukura mienzaniso yakati wandei yematambudziko anosanganisira madenderedzwa nema tangent.
Kunzwisisa Kwekutanga kweMadenderedzwa neMatanho
Denderedzwa
Denderedzwa chimiro chejometri chakaumbwa neboka remapoinzi ese ari mudenderedzwa ari chinhambwe chakatarwa kubva panzvimbo yakatarwa inonzi pakati pedenderedzwa. Dara iri rakatarwa rinonzi radius yedenderedzwa.
Tangent
Tangent kudenderedzwa mutsetse unobata denderedzwa panzvimbo imwe chete chaiyo. Nzvimbo iyi inonzi point of tangency. Tangents dzine zvinhu zvakakosha zvakasiyana-siyana, zvinosanganisira:
– Mutsetse we tangent unogara wakatarisana ne radius yedenderedzwa panzvimbo ye tangency.
– Kureba kwe tangent kubva panzvimbo iri kunze kwedenderedzwa kuenda kudenderedzwa kwakafanana kana tangent mbiri dzakatorwa kubva panzvimbo iyoyo.
Mibvunzo yemuenzaniso nehurukuro
Pazasi tichapa mienzaniso yakati wandei yemibvunzo inokurukura zvakadzama nezvepfungwa yemadenderedzwa nematangenti.
Muenzaniso Mubvunzo 1: Kuwana Kureba kweMutsetse weTangent
Mubvunzo:
Kana wapiwa denderedzwa rine pakati \(O\) uye radius \(r = 6 \, \text{cm}\). Kubva panzvimbo \(P\) kunze kwedenderedzwa iri 10 cm kubva pakati pedenderedzwa, matangenti maviri \(PA\) uye \(PB\) anodhonzwa kudenderedzwa. Verenga kureba kwetangenti \(PA\).
Kukurukurirana:
Mudambudziko iri, tinogona kushandisa dzidziso yePythagorean. Dhirowa triangle \(\triangle OAP\):
– \(OP = 10 \, \text{cm}\) (daro kubva panzvimbo yekunze kusvika pakati pedenderedzwa)
– \(OA = 6 \, \text{cm}\) (nzvimbo yedenderedzwa)
– \(PA\) ndiwo mutsetse we tangent unofanirwa kuwanikwa
\[
OP^2 = OA^2 + PA^2
\]
\[
10^2 = 6^2 + PA^2
\]
\[
100 = 36 + PA^2
\]
\[
PA^2 = 64
\]
\[
PA = \sqrt{64} = 8 \, \text{cm}
\]
Saka, kureba kwemutsetse we tangent \(PA\) i 8 cm.
Muenzaniso Mubvunzo 2: Kuwana Pfungwa Yekutendeseka
Mubvunzo:
Kupiwa denderedzwa rine equation \((x – 3)^2 + (y – 4)^2 = 25\) uye mutsetse \(y = 2x + 1\). Sarudza poindi yekubatana pakati pedenderedzwa nemutsetse.
Kukurukurirana:
Kutanga, tinoona pakati nepakati pedenderedzwa:
– Nzvimbo \(O(3, 4)\)
– Nharaunda \(r = \sqrt{25} = 5\)
Kuti tiwane poindi ye tangency, ngatifungei kuti poindi ye tangency ndi \(T(x_1, y_1)\) iyo iriwo pamutsetse \(y = 2x + 1\). Zvadaro:
\[
y_1 = 2x_1 + 1
\]
\(T(x_1, y_1)\) inofanirawo kuzadzisa equation yedenderedzwa:
\[
(x_1 – 3)^2 + (y_1 – 4)^2 = 25
\]
Isa \(y_1 = 2x_1 + 1\) muequation yedenderedzwa:
\[
(x_1 – 3)^2 + ((2x_1 + 1) – 4)^2 = 25
\]
\[
(x_1 – 3)^2 + (2x_1 – 3)^2 = 25
\]
Tinofanira kuverenga masikweya maviri.
\[
(x_1 – 3)^2 = x_1^2 – 6x_1 + 9
\]
\[
(2x_1 – 3)^2 = 4x_1^2 – 12x_1 + 9
\]
Sanganisa mhedzisiro yese iri miviri:
\[
x_1^2 – 6x_1 + 9 + 4x_1^2 – 12x_1 + 9 = 25
\]
\[
5x_1^2 – 18x_1 + 18 = 25
\]
Bvisa 25 kubva kumativi ese:
\[
5x_1^2 – 18x_1 – 7 = 0
\]
Gadzirisa equation ye quadratic:
\[
x_1 = \frac{18 \pm \sqrt{18^2 + 4 \kawa 5 \kawa 7}}{2 \kawa 5}
\]
\[
x_1 = \frac{18 \pm \sqrt{324 + 140}}{10}
\]
\[
x_1 = \frac{18 \pm \sqrt{464}}{10}
\]
\[
x_1 = \frac{18 \pm 2\sqrt{116}}{10}
\]
\[
x_1 = \frac{18 \pm 2\sqrt{4 \kawa 29}}{10}
\]
\[
x_1 = \frac{18 \pm 4\sqrt{29}}{10}
\]
\[
x_1 = 1.8 \pm 0.4\sqrt{29}
\]
Verenga kukosha kwe \(y_1\):
Izvo zvinogutsa y = 2x + 1:
– Kana \(x_1 = 1.8 + 0.4\sqrt{29}\), saka \(y_1 = 2(1.8 + 0.4\sqrt{29}) + 1\)
– Kana \(x_1 = 1.8 – 0.4\sqrt{29}\), saka \(y_1 = 2(1.8 – 0.4\sqrt{29}) + 1\)
Kuongorora:
Saka tinowana mapoinzi maviri ekusangana kwe equation yedenderedzwa nemutsetse iwoyo.
Muenzaniso Mubvunzo 3: Kusarudza Equation yeTangent Line
Mubvunzo:
Kupiwa denderedzwa rine equation \((x – 2)^2 + (y – 3)^2 = 20\). Sarudza equation yemutsetse wetangent kune denderedzwa rinopfuura nepakati pepoindi \((6, 7)\).
Kukurukurirana:
Chinotenderera kudenderedzwa rine pakati \((h, k)\) uye radius \(r\) kubva panzvimbo yekunze inozivikanwa chinogona kuwanikwa ne equation:
Mutsetse we tangent unopfuura nepanzvimbo yekunze \((x_1, y_1)\):
\[
(x – 2)(x_1 – 2) + (y – 3)(y_1 – 3) = 20
\]
Tsiva poindi yekunze \((6, 7)\):
\[
(x – 2) (6 – 2) + (y – 3) (7 – 3) = 20
\]
\[
4(x – 2) + 4(y – 3) = 20
\]
\[
4(x – 2 + y – 3) = 20
\]
\[
4x + 2y -20 = 20
\]
\[
4x + 4y -20 = 20
\]
\[
x +y = 5
\]
Equation yemutsetse wetangent ndeiyi:
\[
x +y = 9
\]
Saka, musiyano we equation yemutsetse unopfuura nepakati pemutsetse wedenderedzwa wakakura zvikuru uye unogona kuchinja zvichienderana nemhedzisiro kana mufananidzo unoonekwa.
Mhedziso
Kukurukurirana kwemadenderedzwa nematangenti kunosanganisira zvinhu zvakakosha zvemasvomhu, kubva pakushandisa mafomura ekutanga senge dzidziso yePythagorean kusvika pakugadzirisa maequation equadratic. Kuburikidza nemienzaniso iyi, tinogona kunzwisisa zviri nani mashandisirwo epfungwa idzi mumamiriro ezvinhu akaoma. Tinovimba kuti chinyorwa chino chakabatsira kupa mufananidzo wakajeka wekuti tingagadzirisa sei matambudziko anosanganisira madenderedzwa nematangenti.
