Muenzaniso weMubvunzo weKukurukurirana paEquation yeDenderedzwa
Kuenzana kwedenderedzwa inyaya inokosha mukuongorora maekiyumu. Kunzwisisa zvakanaka equation yedenderedzwa kunobatsira zvikuru, kwete mumasvomhu chete asiwo mumashandisirwo akasiyana-siyana einjiniya nesainzi. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yemaequation edenderedzwa nemhinduro dzawo. Chinangwa ndechekupa pfupiso yakajeka uye yakazara yekuti tingagadzirisa sei matambudziko ane chekuita nemaequation edenderedzwa.
Equation Yese yeDenderedzwa
Equation inonyanya kutaurwa yedenderedzwa muCartesian coordinates ndeiyi:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
Di mana:
– \( (a, b) \) ndiwo makoronesheni ari pakati pedenderedzwa.
– \( r \) ndiyo dhayamita yedenderedzwa.
Kana pakati pedenderedzwa pari panzvimbo \( (0, 0) \), equation yedenderedzwa ichave:
\[ x^2 + y^2 = r^2 \]
Zvino, ngatikurukurei mimwe mienzaniso yemibvunzo nemhinduro dzayo.
Muenzaniso Mubvunzo 1
Mubvunzo: Sarudza equation yedenderedzwa rine pakati paro riri panzvimbo (3, -2) uye rine radius ye5.
Mhinduro:
Shandisa fomura yakajairika yekuenzanisa denderedzwa:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
Tsiva ma values \( a = 3 \), \( b = -2 \), uye \( r = 5 \):
\[ (x – 3)^2 + (y + 2)^2 = 5^2 \]
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Saka, equation yedenderedzwa ndeiyi:
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Muenzaniso Mubvunzo 2
Mubvunzo: Sarudza equation yedenderedzwa rine pakati paro pari pakutanga (0, 0) uye rine radius ye7.
Mhinduro:
Sezvo pakati pedenderedzwa pari pakutanga, tinogona kushandisa equation iri nyore:
\[ x^2 + y^2 = r^2 \]
Tsiva kukosha \( r = 7 \):
\[ x^2 + y^2 = 7^2 \]
\[ x^2 + y^2 = 49 \]
Saka, equation yedenderedzwa ndeiyi:
\[ x^2 + y^2 = 49 \]
Muenzaniso Mubvunzo 3
Mubvunzo: Sarudza equation yedenderedzwa rine pakati paro (4, -5) uye rinobata Y axis.
Mhinduro:
Denderedzwa rakatenderedzwa neY-axis zvinoreva kuti daro kubva pakati pedenderedzwa kusvika kuY-axis rakaenzana neradius yaro. Daro iri ndiro kukosha kwakakwana kweX-coordinate yepakati pedenderedzwa. Saka, radius i4.
Shandisa fomura yakajairika yekuenzanisa denderedzwa:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
Tsiva ma values \( a = 4 \), \( b = -5 \), uye \( r = 4 \):
\[ (x – 4)^2 + (y + 5)^2 = 4^2 \]
\[ (x – 4)^2 + (y + 5)^2 = 16 \]
Saka, equation yedenderedzwa ndeiyi:
\[ (x – 4)^2 + (y + 5)^2 = 16 \]
Muenzaniso Mubvunzo 4
Mubvunzo: Denderedzwa rine equation \( x^2 + y^2 – 6x + 4y – 12 = 0 \). Sarudza pakati nepakati pedenderedzwa.
Mhinduro:
Kuti tigadzirise equation iyi, tinofanira kuishandura kuita standard form \( (x – a)^2 + (y – b)^2 = r^2 \). Matanho ekuipedzisa ndeaya anotevera:
1. Kuronga nekugadzirisa masikweya akakwana:
Equation yekutanga ndeiyi:
\[ x^2 + y^2 – 6x + 4y – 12 = 0 \]
Boka \( x \) uye \( y \):
\[ (x^2 – 6x) + (y^2 + 4y) = 12 \]
2. Gadzirisa sikweya yakakwana:
Kune \( x^2 – 6x \):
\[ x^2 – 6x + 9 \]
Kune \( y^2 + 4y \):
\[ y^2 + 4y + 4 \]
Wedzera 9 ne4 kumativi ese e equation:
\[ (x^2 – 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4 \]
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Saka, equation yedenderedzwa muchimiro chakajairika ndeiyi:
\[ (x – 3)^2 + (y + 2)^2 = 25 \]
Kubva pano, tinogona kuona kuti pakati pedenderedzwa ndi \( (3, -2) \) uye radius ndi \( r = \sqrt{25} = 5 \).
Muenzaniso Mubvunzo 5
Mubvunzo: Sarudza equation yedenderedzwa inopfuura nepakati pemapoinzi (2, 3) uye (4, 5), uye pakati payo pamutsetse x = 3.
Mhinduro:
Kubva pamubvunzo, tinoziva kuti pakati pedenderedzwa ndi (3, b). Denderedzwa rinopfuurawo nepakati penzvimbo mbiri dzinozivikanwa. Sezvo denderedzwa richipfuura nepakati pa (2, 3), daro kubva pakati kusvika pano ndiro radius.
Equation yedenderedzwa ndeiyi:
\[ (x – 3)^2 + (y – b)^2 = r^2 \]
Nzvimbo yekutsiva (2, 3):
\[ (2 – 3)^2 + (3 – b)^2 = r^2 \]
\[ 1 + (3 – b)^2 = r^2 \]
\[ (3 – b)^2 = r^2 – 1 \]
Nzvimbo yekutsiva (4, 5):
\[ (4 – 3)^2 + (5 – b)^2 = r^2 \]
\[ 1 + (5 – b)^2 = r^2 \]
\[ (5 – b)^2 = r^2 – 1 \]
Kubva muequation mbiri idzi, tinoziva (3 - b)^2 = (5 - b)^2. Saka:
\[ 3 – b = \pm(5 – b) \]
Kana \( 3 – b = 5 – b \), mhedzisiro yacho haigone kuva yechokwadi. Saka:
\[ 3 – b = -(5 – b) \]
\[b = 4 \]
Kana b = 4, equation yedenderedzwa ndeiyi:
\[ (x – 3)^2 + (y – 4)^2 = 2 \]
Zvisinei, tinogona kuverenga radius r kubva padaro riri pakati pepakati nepoindi (2, 3) = \(\sqrt{(2 – 3)^2 + (3 – 4)^2} \) = \(\sqrt{1+1}\) = \(\sqrt {2}\)
Equation yedenderedzwa ndeiyi:
\[ (x – 3)^2 + (y – 4)^2 = 2 \]
Mhedziso
Kunzwisisa equation yedenderedzwa kunogona kuita kuti kugadzirisa matambudziko mazhinji emasvomhu kuve nyore. Muchiitiko chimwe nechimwe, kuziva pakati nepakati penzira kwakakosha. Tinovimba kuti matambudziko aya emuenzaniso netsananguro dzawo zvinopa kujekesa uye zvinokubatsira kudzidza equation yedenderedzwa. Kudzidzira kunoita kuti zvive zvakanaka mu masvomhu, saka usazeza kuedza matambudziko akasiyana-siyana kuti uvandudze hunyanzvi hwako.
