Antes de ver o que \u00e9 a equa\u00e7\u00e3o da circunfer\u00eancia, vamos relembrar a no\u00e7\u00e3o de circunfer\u00eancia:<\/p>\n
A circunfer\u00eancia \u00e9 o lugar geom\u00e9trico dos pontos do plano equidistantes de um ponto fixo denominado centro. <\/p>\n Portanto, todos os pontos de um c\u00edrculo est\u00e3o \u00e0 mesma dist\u00e2ncia do seu centro.<\/p>\n Al\u00e9m disso, o c\u00edrculo \u00e9 uma das quatro se\u00e7\u00f5es c\u00f4nicas junto com a elipse, a par\u00e1bola e a hip\u00e9rbole. Ou seja, um c\u00edrculo pode ser obtido cortando-se um cone com um plano paralelo \u00e0 sua base.<\/p>\n A maneira mais simples de descrever um c\u00edrculo no plano cartesiano \u00e9 a partir de sua equa\u00e7\u00e3o ordin\u00e1ria. Portanto, a f\u00f3rmula para a equa\u00e7\u00e3o ordin\u00e1ria da circunfer\u00eancia \u00e9 a seguinte: <\/p>\n A equa\u00e7\u00e3o ordin\u00e1ria do c\u00edrculo<\/strong> \u00e9:<\/p>\n<\/p>\n Ouro:<\/p>\n \u00e9 o raio do c\u00edrculo.<\/li>\n E<\/p>\n s\u00e3o as coordenadas do centro do c\u00edrculo:<\/p>\n Embora n\u00e3o a demonstremos por ser um pouco tedioso, esta equa\u00e7\u00e3o pode ser obtida a partir do teorema de Pit\u00e1goras.<\/p>\n Vamos ver como a equa\u00e7\u00e3o ordin\u00e1ria de um c\u00edrculo \u00e9 calculada com um exemplo:<\/p>\n A f\u00f3rmula para a equa\u00e7\u00e3o ordin\u00e1ria de um c\u00edrculo \u00e9:<\/p>\n<\/p>\n Portanto, s\u00f3 temos que substituir a inc\u00f3gnita<\/p>\n<\/p>\n pelo valor do raio, e as inc\u00f3gnitas<\/p>\n E<\/p>\n pelas coordenadas X e Y respectivamente do centro do c\u00edrculo:<\/p>\n<\/p>\n Portanto, a equa\u00e7\u00e3o ordin\u00e1ria da circunfer\u00eancia \u00e9: <\/p>\n<\/p>\n Outro tipo de equa\u00e7\u00e3o de circunfer\u00eancia \u00e9 a equa\u00e7\u00e3o geral, ali\u00e1s \u00e9 a mais utilizada. Veremos ent\u00e3o como obter a equa\u00e7\u00e3o geral de qualquer circunfer\u00eancia a partir da sua equa\u00e7\u00e3o ordin\u00e1ria.<\/p>\n Considere a equa\u00e7\u00e3o ordin\u00e1ria de um c\u00edrculo:<\/p>\n<\/p>\n Se desenvolvermos as igualdades not\u00e1veis (ou produtos not\u00e1veis):<\/p>\n<\/p>\n Agora fazemos 3 altera\u00e7\u00f5es de vari\u00e1veis:<\/p>\n<\/p>\n E finalmente obtemos a equa\u00e7\u00e3o geral da circunfer\u00eancia: <\/p>\n<\/p>\n A f\u00f3rmula para a equa\u00e7\u00e3o geral da circunfer\u00eancia<\/strong> \u00e9, portanto:<\/p>\n<\/p>\n onde o centro do c\u00edrculo \u00e9:<\/p>\n<\/p>\n E o raio do c\u00edrculo \u00e9:<\/p>\n<\/p>\n Portanto, esta equa\u00e7\u00e3o da circunfer\u00eancia \u00e9 sempre obtida pela equa\u00e7\u00e3o ordin\u00e1ria. Aqui est\u00e1 um exemplo para ver como isso \u00e9 feito:<\/p>\n Primeiro precisamos encontrar a equa\u00e7\u00e3o ordin\u00e1ria da circunfer\u00eancia. Para fazer isso, usamos sua f\u00f3rmula:<\/p>\n<\/p>\n E agora operamos at\u00e9 encontrarmos a equa\u00e7\u00e3o geral da circunfer\u00eancia, ou seja, at\u00e9 n\u00e3o podermos mais simplificar: <\/p>\n<\/p>\n Portanto a equa\u00e7\u00e3o geral da circunfer\u00eancia \u00e9:<\/p>\n<\/p>\n Embora o problema n\u00e3o exigisse isso, podemos agora calcular o centro e o raio da equa\u00e7\u00e3o encontrada para verificar se est\u00e1 correta.<\/p>\n Para determinar o centro do c\u00edrculo, usamos sua f\u00f3rmula: <\/p>\n<\/p>\n Na verdade, o centro do c\u00edrculo coincide com o da afirma\u00e7\u00e3o.<\/p>\n Tamb\u00e9m verificamos o raio da circunfer\u00eancia com sua f\u00f3rmula:<\/p>\n<\/p>\n E o raio tamb\u00e9m \u00e9 igual ao da afirma\u00e7\u00e3o. Portanto, a equa\u00e7\u00e3o da circunfer\u00eancia calculada est\u00e1 correta.<\/p>\n Todas as equa\u00e7\u00f5es na forma de<\/p>\n<\/p>\n correspondem a um c\u00edrculo. Assim, para que este tipo de express\u00e3o seja verdadeiramente a equa\u00e7\u00e3o de um c\u00edrculo, as 3 condi\u00e7\u00f5es a seguir devem ser atendidas:<\/p>\n <\/span> e de<\/p>\n eles devem ser iguais a 1. Lembre-se de que se ambas as vari\u00e1veis fossem precedidas por um n\u00famero diferente de um, mas ambas tivessem o mesmo n\u00famero, toda a equa\u00e7\u00e3o poderia ser dividida por esse n\u00famero para que seus coeficientes fossem 1.<\/li>\n <\/span><\/li>\n As duas equa\u00e7\u00f5es circulares que vimos, a equa\u00e7\u00e3o ordin\u00e1ria e a equa\u00e7\u00e3o geral, s\u00e3o as mais utilizadas para expressar matematicamente uma circunfer\u00eancia no plano (em R2). Por\u00e9m, existem v\u00e1rios tipos de equa\u00e7\u00f5es para descrever este objeto geom\u00e9trico, abaixo est\u00e1 a explica\u00e7\u00e3o de cada uma delas. <\/p>\n A equa\u00e7\u00e3o can\u00f4nica, ou equa\u00e7\u00e3o reduzida, de um c\u00edrculo \u00e9 usada para descrever qualquer c\u00edrculo cujo centro est\u00e1 na origem das coordenadas<\/strong> , ou seja, no ponto (0,0). A referida equa\u00e7\u00e3o \u00e9 a seguinte:<\/p>\n<\/p>\n Se, al\u00e9m disso, o raio fosse equivalente \u00e0 unidade (1), a equa\u00e7\u00e3o da circunfer\u00eancia seria:<\/p>\n<\/p>\n Esta \u00faltima equa\u00e7\u00e3o corresponde \u00e0 circunfer\u00eancia goniom\u00e9trica, tamb\u00e9m chamada de circunfer\u00eancia unit\u00e1ria ou c\u00edrculo unit\u00e1rio. \u00c9 o c\u00edrculo de raio 1 centrado na origem das coordenadas. <\/p>\n Duas equa\u00e7\u00f5es conc\u00eantricas s\u00e3o aquelas que t\u00eam o centro no mesmo ponto. E a \u00fanica coisa diferente que dois c\u00edrculos conc\u00eantricos t\u00eam \u00e9 o raio.<\/p>\n Portanto, para que esta condi\u00e7\u00e3o seja satisfeita, as equa\u00e7\u00f5es de dois c\u00edrculos conc\u00eantricos s\u00e3o exatamente iguais, exceto pelos seus termos independentes, que devem ser diferentes.<\/strong><\/p>\n<\/p>\n Por exemplo, os dois c\u00edrculos seguintes s\u00e3o conc\u00eantricos, uma vez que todos os seus coeficientes s\u00e3o id\u00eanticos, exceto os termos independentes: <\/p>\n<\/p>\n Assim como a reta, a equa\u00e7\u00e3o de um c\u00edrculo tamb\u00e9m pode ser parametrizada com as fun\u00e7\u00f5es trigonom\u00e9tricas de seno e cosseno. Assim, as equa\u00e7\u00f5es param\u00e9tricas do c\u00edrculo s\u00e3o:<\/p>\n<\/p>\n onde o ponto<\/p>\n<\/p>\n \u00e9 o centro do c\u00edrculo e<\/p>\n Este \u00e9 o seu departamento. <\/p>\n Calcule a equa\u00e7\u00e3o geral do c\u00edrculo de raio 5 cujo centro est\u00e1 no ponto <\/p>\n<\/p>\n Para determinar a equa\u00e7\u00e3o geral da circunfer\u00eancia, devemos primeiro determinar a sua equa\u00e7\u00e3o ordin\u00e1ria. Para fazer isso, usamos a f\u00f3rmula da equa\u00e7\u00e3o ordin\u00e1ria de um c\u00edrculo: <\/p>\n<\/p>\n Uma vez conhecida a equa\u00e7\u00e3o ordin\u00e1ria, trabalhamos at\u00e9 encontrar a equa\u00e7\u00e3o geral da circunfer\u00eancia: <\/p>\n<\/p>\n Portanto a equa\u00e7\u00e3o geral da circunfer\u00eancia \u00e9: <\/p>\n<\/p>\n Para cada um dos c\u00edrculos a seguir, encontre as coordenadas do seu centro e o comprimento do seu raio. <\/p>\n<\/p>\n circunfer\u00eancia A)<\/strong><\/p>\n<\/p>\n A circunfer\u00eancia \u00e9 expressa na forma de uma equa\u00e7\u00e3o ordin\u00e1ria, cuja f\u00f3rmula \u00e9:<\/p>\n<\/p>\n Portanto, as coordenadas do centro do c\u00edrculo s\u00e3o: <\/p>\n<\/p>\n E seu raio \u00e9: <\/p>\n<\/p>\n circunfer\u00eancia B)<\/strong><\/p>\n<\/p>\n Esta circunfer\u00eancia \u00e9 expressa na forma de uma equa\u00e7\u00e3o geral, portanto para calcular as coordenadas do seu centro deve-se utilizar a seguinte f\u00f3rmula: <\/p>\n<\/p>\n Por outro lado, a f\u00f3rmula para encontrar o raio do c\u00edrculo \u00e9:<\/p>\n<\/p>\n circunfer\u00eancia C)<\/strong><\/p>\n<\/p>\n A circunfer\u00eancia \u00e9 expressa na forma de uma equa\u00e7\u00e3o ordin\u00e1ria, cuja f\u00f3rmula \u00e9:<\/p>\n<\/p>\n Portanto, as coordenadas do centro do c\u00edrculo s\u00e3o:<\/p>\n<\/p>\n Neste caso, a equa\u00e7\u00e3o n\u00e3o tem termo.<\/p>\n<\/p>\n nenhum<\/p>\n est\u00e1, portanto, centrado na origem das coordenadas:<\/p>\n<\/p>\n E seu raio \u00e9: <\/p>\n<\/p>\n Qual das seguintes equa\u00e7\u00f5es \u00e9 a equa\u00e7\u00e3o de um c\u00edrculo? <\/p>\n<\/p>\n Para que uma express\u00e3o seja a equa\u00e7\u00e3o de um c\u00edrculo, as seguintes condi\u00e7\u00f5es devem ser verdadeiras:<\/p>\n 1.<\/strong> Os coeficientes de<\/p>\n<\/p>\n e de<\/p>\n Eles devem ser iguais a 1. 3.<\/strong><\/p>\n Devemos, portanto, verificar se as tr\u00eas condi\u00e7\u00f5es s\u00e3o atendidas para cada equa\u00e7\u00e3o. <\/p>\n Equa\u00e7\u00e3o A)<\/strong><\/p>\n<\/p>\n Os coeficientes de<\/p>\n<\/p>\n E<\/p>\n s\u00e3o 1 e a equa\u00e7\u00e3o n\u00e3o tem termo<\/p>\n Basta, portanto, verificar a terceira condi\u00e7\u00e3o: <\/p>\n<\/p>\n A equa\u00e7\u00e3o satisfaz as 3 condi\u00e7\u00f5es, portanto \u00e9 a equa\u00e7\u00e3o de um c\u00edrculo.<\/strong> <\/p>\n Equa\u00e7\u00e3o B)<\/strong><\/p>\n<\/p>\n A equa\u00e7\u00e3o tem um termo com<\/p>\n<\/p>\n com o qual a equa\u00e7\u00e3o n\u00e3o corresponde a um c\u00edrculo.<\/strong> <\/p>\n Equa\u00e7\u00e3o C)<\/strong><\/p>\n<\/p>\n Os coeficientes de<\/p>\n<\/p>\n E<\/p>\n n\u00e3o s\u00e3o 1, mas podemos transformar a equa\u00e7\u00e3o dividindo todos os termos:<\/p>\n<\/p>\n Desta forma agora os coeficientes de<\/p>\n<\/p>\n E<\/p>\n sim, eles s\u00e3o 1 e, al\u00e9m disso, a equa\u00e7\u00e3o n\u00e3o tem termo<\/p>\n Portanto, s\u00f3 temos que corroborar a terceira condi\u00e7\u00e3o: <\/p>\n<\/p>\n A equa\u00e7\u00e3o satisfaz as 3 condi\u00e7\u00f5es, portanto \u00e9 a equa\u00e7\u00e3o de um c\u00edrculo.<\/strong> <\/p>\n Equa\u00e7\u00e3o D)<\/strong><\/p>\n<\/p>\n Os coeficientes de<\/p>\n<\/p>\n E<\/p>\n s\u00e3o 1 e a equa\u00e7\u00e3o n\u00e3o tem termo<\/p>\n Basta, portanto, verificar a terceira condi\u00e7\u00e3o: <\/p>\n<\/p>\n A equa\u00e7\u00e3o n\u00e3o satisfaz a \u00faltima condi\u00e7\u00e3o, portanto n\u00e3o \u00e9 a equa\u00e7\u00e3o de um c\u00edrculo<\/strong> .<\/p>\n Determine a equa\u00e7\u00e3o da circunfer\u00eancia que passa pelos tr\u00eas pontos a seguir: <\/p>\n<\/p>\n A equa\u00e7\u00e3o geral de qualquer c\u00edrculo \u00e9:<\/p>\n<\/p>\n Portanto, precisamos substituir as coordenadas dos pontos na equa\u00e7\u00e3o do c\u00edrculo para encontrar os par\u00e2metros<\/p>\n<\/p>\n E<\/p>\n Com o primeiro ponto de encontramos o coeficiente <\/p>\n<\/p>\n Com o segundo ponto encontramos o coeficiente <\/p>\n<\/p>\n E a partir do terceiro ponto encontramos o coeficiente <\/p>\n<\/p>\n Em conclus\u00e3o, a equa\u00e7\u00e3o geral para circunfer\u00eancia \u00e9:<\/p>\n<\/p>\n Se as extremidades opostas de um c\u00edrculo s\u00e3o os dois pontos a seguir:<\/p>\n<\/p>\n Qual \u00e9 a equa\u00e7\u00e3o ordin\u00e1ria do c\u00edrculo? <\/p>\n Se os dois pontos forem extremos do c\u00edrculo, seu centro ser\u00e1 o ponto m\u00e9dio entre esses dois pontos: <\/p>\n<\/p>\n Por outro lado, o di\u00e2metro do c\u00edrculo ser\u00e1 a dist\u00e2ncia entre os dois pontos, que pode ser calculada utilizando a magnitude do vetor que os dois pontos formam: <\/p>\n<\/p>\n E o raio do c\u00edrculo \u00e9 metade do di\u00e2metro:<\/p>\n<\/p>\n A equa\u00e7\u00e3o ordin\u00e1ria do c\u00edrculo \u00e9, portanto: <\/p>\n<\/p>\n Por fim, se este artigo foi \u00fatil para voc\u00ea, certamente tamb\u00e9m se interessar\u00e1 por nossas p\u00e1ginas de hip\u00e9rbole (matem\u00e1tica)<\/a> e par\u00e1bola (matem\u00e1tica)<\/a> . Voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o detalhada do que s\u00e3o a hip\u00e9rbole e a par\u00e1bola, suas equa\u00e7\u00f5es, suas caracter\u00edsticas, exemplos, exerc\u00edcios resolvidos\u2026<\/p>\n","protected":false},"excerpt":{"rendered":" Nesta p\u00e1gina voc\u00ea encontrar\u00e1 tudo sobre a equa\u00e7\u00e3o da circunfer\u00eancia: equa\u00e7\u00e3o ordin\u00e1ria, equa\u00e7\u00e3o geral, outros tipos de equa\u00e7\u00f5es da circunfer\u00eancia, quando a equa\u00e7\u00e3o de uma circunfer\u00eancia est\u00e1 correta,\u2026 Al\u00e9m disso, voc\u00ea ver\u00e1 exemplos de como encontrar o equa\u00e7\u00e3o de uma circunfer\u00eancia e voc\u00ea pode praticar com exerc\u00edcios resolvidos. Equa\u00e7\u00e3o ordin\u00e1ria do c\u00edrculo Antes de ver …<\/p>\n![\"qual](\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equation-de-circonference-1.webp\")
<\/figure>\n![\"(x-a)^2+(y-b)^2=r^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb8ffc83b27f7ed251e2233abbce5046_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n\n
![\"r\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" "\\"Rendered")
<\/p>\n![\"a\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" "\\"Rendered")
<\/p>\n![\"b\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" "\\"Rendered")
<\/p>\n![\"C(a,b)\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3f5efef3ea46a765419553313f5ca6ec_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n<\/ul>\n<\/div>\n\n
![\"C(3,-1).\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61ed8ca74441b9c9ed7efef1a47bd204_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n<\/ul>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"(x-3)^2+(y-(-1))^2=5^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b3bfb9b8a5a2dfb5d73ed3986567c4f_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\bm{(x-3)^2+(y+1)^2=25}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5847df3cfa46d4ba66d0caba4026dabe_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Equa\u00e7\u00e3o geral do c\u00edrculo<\/span><\/h2>\n
<\/p>\n<\/p>\n![\"x^2+a^2-2ax+y^2+b^2-2by=r^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a4d2b00fd17214927aa165291c8aebce_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2-2ax+y^2-2by+a^2+b^2-r^2=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f787480794d72c03ba5880758dedeee_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"A=-2a](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f0414589c97ababc91d90e8b4d76f3c_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+y^2+Ax+By+C=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8890b7cc400fafc2f805ca67e08881ea_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e64305413efd35fdc4d0650acf78623d_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee6df58c142869c04d0db23bda17b0fc_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/div>\n\n
![\"C(2,4).\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-39163542e992aade309d98d420e1a89e_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n<\/ul>\n<\/p>\n<\/p>\n![\"(x-2)^2+(y-4)^2=6^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b24acf89d8fb3cfb6b48d8ce985f3e2_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+2^2-2\\cdot](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68b827aa8155ab22926ac9ab1ae78745_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+4-4x+y^2+16-8y=36\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6651ecf7c153c784e8a0c7e01522be4_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2-4x+y^2-8y+4+16-36=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e0dcac8662f04c52e566c31069c1615_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2-4x+y^2-8y-16=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e73476ebd4ceb89a5683bcb06af09eda_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\bm{x^2+y^2-4x-8y-16=0}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdb5e1831804822235795e661323b911_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a0eb46f8005ca0cb4418d019c01d090b_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8350046e2eed3457177b5d4c040fcdcf_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fa2b1a6a7b85d8f05bf091b50d99e060_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-04dc99d462a8ceb5e6dca936999a4ce8_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Exist\u00eancia de uma circunfer\u00eancia<\/span><\/h2>\n
<\/p>\n\n
![\"x^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76e092d71026e8d64e9e3fc6857554cc_l3.png\" "\\"Rendered")
<\/p>\n![\"y^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c73c29a3e05add4949b9207310200e96_l3.png\" "\\"Rendered")
<\/p>\n![\"xy.\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce7d7fd003c1b94cfc8c5c46f148f714_l3.png\" "\\"Rendered")
<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ffc789338787de266be2abc66e24a455_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”190″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n<\/ol>\n<\/span> Outros tipos de equa\u00e7\u00f5es circulares<\/span><\/h2>\n
<\/span> Equa\u00e7\u00e3o can\u00f4nica do c\u00edrculo<\/span><\/h3>\n
![\"x^2+y^2=r^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8837a5c798c950520f838adb2826aa8_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+y^2=1\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6638c5ad97a435df56133e8b04c26ff4_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Equa\u00e7\u00f5es de dois c\u00edrculos conc\u00eantricos<\/span><\/h3>\n
<\/p>\n<\/p>\n![\"x^2+y^2+Ax+By+C'=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c0ebfea4141df270980aa34c849d6cfa_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+y^2+3x-4y+1=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7da3e573efeee0a54f17639bb938d787_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+y^2+3x-4y+5=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e569a0908d1a2537bf5dd0d5fcaa76bc_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Equa\u00e7\u00e3o param\u00e9trica do c\u00edrculo<\/span><\/h3>\n
![\"\\diplaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72dae5b3c5f775661ecb102945bb3fbd_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"(a,b)\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe11303e9d78d9884d5ce80743b4e853_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Problemas resolvidos da equa\u00e7\u00e3o de um c\u00edrculo<\/span><\/h2>\n
Exerc\u00edcio 1<\/h3>\n
![\"C(-1,2).\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5fa41d0801e98e2a7874c963fff73d9_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"(x-(-1))^2+(y-2)^2=5^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6aaa336cc3cb4df0e6bfe594bb6e7d22_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"(x+1)^2+(y-2)^2=25\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2647ce60ac9f3b4bddf17ded227da332_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+1^2+2\\cdot](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dacb70230bf38f0719ef399c3bf97727_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+1+2x+y^2+4-4y=25\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c540e082018d26478d17e7cd1991280e_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+2x+y^2-4y+1+4-25=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a84db66d6be46dcd6577bc788051cc6f_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+2x+y^2-4y-20=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61dbc21597fd549b0826f81395467ae8_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\bm{x^2+y^2+2x-4y-20=0}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d0aad138d023e75ed59b3fa7b89247ea_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\nExerc\u00edcio 2<\/h3>\n
![\"\\text{A)}\\](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-93caa68e4fb4ec47816d1d8e6bf3a406_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\text{B)}\\](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb7951cbb0d649bd68e0a7be0d66281c_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\text{C)}\\](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b63bd92988f9ebb0e256e7b6f72120e4_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"(x-2)^2+(y+5)^2=36\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8c045e659d071fcc7f64053f6be107f_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"\\bm{C(2,-5)}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aeae2f2253b0e865affe7eb4e4045c5a_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"r^2=36\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-600a7310c5834d8d3fad25e89deba0a1_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\bm{r=6}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9233e539f37236d7548131ed75ad1e86_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n
\n![\"x^2+y^2+8x-10y+1](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-253fa91392e1e066a52be943874175cf_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-554c02287c317db9bfad3ee656ffc4d6_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e07d270b9ef4a798ab46911132c60bf0_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9eb4c6f0c580fa8cd0304d95339f1e49_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0cdc38937e6c36bb9e6d2ca0814a650_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"x^2+y^2=4\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91652bf524a31b3abfbacf829d5c3024_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"b,\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a521efcd0a946cd643aebe98b5b41a3c_l3.png\" "\\"Rendered")
<\/p>\n![\"\\bm{C(0,0)}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-af6eeb157115e2a18cfa38f09fc11d85_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"r^2=4\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f15f619bc019d0f668cff042ed62ed6_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\bm{r=2}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4964d8cfd7ef22134e4d600697e4841d_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\nExerc\u00edcio 3<\/h3>\n
![\"\\text{A)}\\](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-669f1a6230f3055c7cb7a40db679ef48_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\text{B)}\\](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef04a07d4a0089b39ecbc9de3047c185_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\text{C)}\\](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bdd38651c9b1ac5cbfc2315da374ead6_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\text{D)}\\](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f222b1efd24b2c061b293b5d0d12a374_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n
2.<\/strong> A equa\u00e7\u00e3o n\u00e3o pode ter termo<\/p>\n<\/p>\n0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”190″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n
\n![\"x^2+y^2+4x-6y-1=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a6a81516c0f8432b89f8bc6f90ac9812_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”190″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48023f09730538bbe1c444ce95734453_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”216″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5ef2259d5b4e8e18842abbb767a7b293_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”22″ width=”146″ style=”vertical-align: -5px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59ef896ffd96e877fbf3198732f2d3af_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”14″ width=”103″ style=”vertical-align: -2px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91029a12fff0b2f34276f8692b5173a1_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”14″ width=”49″ style=”vertical-align: -2px;”><\/p>\n<\/p>\n
\n![\"x^2+y^2+5x+5y+2xy-4](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326811b6b97f9ba0bc878dfcf8904b21_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"xy,\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a0b5b65b4fa8aef2ecb83e4724b75cc6_l3.png\" "\\"Rendered")
<\/p>\n
\n![\"2x^2+2y^2-8x+4y+2=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7997d025687293da234c60efbb4390ad_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"x^2+y^2-4x+2y+1=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-324f0e4c339c56376f79a6f8420fcb03_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”190″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-283fb580944e951ece8b8e24e84eb524_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”189″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91f0622656a691a59a0e9052709d21de_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”22″ width=”158″ style=”vertical-align: -5px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5eadfd980efe34932b4329383d30563a_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”14″ width=”103″ style=”vertical-align: -2px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c570dbdfdce26cd5e1e13bd563e2ccd_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”14″ width=”42″ style=”vertical-align: -2px;”><\/p>\n<\/p>\n
\n![\"x^2+y^2+x+2y+6=0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b896c6512c871afb6136853a0a2cf72b_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”190″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df508b8ad2882f89d197f3845bcc13ee_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”47″ width=”175″ style=”vertical-align: -17px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06600177f85089d43db1d30e47297e6c_l3.png\")
0″ title=”Rendered by QuickLaTeX.com” height=”17″ width=”128″ style=”vertical-align: -4px;”><\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-74d079e75c8619832a03cd179a3383ce_l3.png\")
} \\ 0″ title=”Rendered by QuickLaTeX.com” height=”17″ width=”82″ style=”vertical-align: -4px;”><\/p>\n<\/p>\n Exerc\u00edcio 4<\/h3>\n
![\"A(0,0)](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4502f53d3453a32786375e0583cb2785_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"A,\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-94b9e9013dcf92b4ae2a6fbb804365a8_l3.png\" "\\"Rendered")
<\/p>\n![\"B\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" "\\"Rendered")
<\/p>\n![\"C.\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e33ac8af21963090f8229a332d211273_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"C:\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab5bb26176c859b9b09d69eb399bea9c_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"A(0,0)](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c994d016e5b89d5820060efa138860cc_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"A:\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-944477c7f7578892a57aa3b7c7dd8268_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\begin{aligned}A(3,0)](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12719654d98ba81f0a7d45e587478e6d_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"B:\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2976704f0327609d6a0deba6d1139c3f_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\begin{aligned}](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1e69c28996a8fbfa14189c48a295095_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\bm{x^2+y^2-3x+y=0}\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9f54f4187497c1fa02459870f4738c3_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\nExerc\u00edcio 5<\/h3>\n
![\"A(2,3)](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1dad063e5dabd63e9833d42ad087a2ce_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8301618bb0d7b4ded6acd02f1087b9e_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe41a577fe933b35601afe7f6239e03d_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06be238c032f258db6686b8e2c164605_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\vv{AB}](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a3dd2c628d11818ba0cab76f4614ed91_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"d](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d0fb498083e41cf9f0f8b7a150f42fd_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"r](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d5a6269d58490fc6848303c3654a1c7_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a958c8b86fa29064dff3aa581e722473_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"\\displaystyle](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-34bc4dc9c86ca1b78edaa23113e51a59_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n