Urgente....................................................................................................................
1) determine o centro e o raio das circunferências de equações dadas.
a) x²-5x+y²-11/4=0
b) x²+y²+ X -6y - 15/4=0
2) verifique se a equação dada representa uma circunferência. em caso afirmativo, determine seus raio e centro.
a) 9x² + 9y² + 6x - 36y + 64=0
b) x² + y² - 3y=7/4
3) determine as retas de declive m= 2 que são tangentes à circunferência x² + y²=5 e faça o gráfico
Soluções para a tarefa
Respondido por
2
Boa tarde Daphinne!
Solução!
Para encontrar o centro de uma circunferencia basta comparar.
![A)\\\\\
x^{2} -5x+ y^{2}- \frac{11}{4}=0\\\\\
x^{2} -5x+ y^{2}= \frac{11}{4}\\\\\
(x- \frac{5}{2})^{2}+(y+0)^{2}= \frac{11}{4}\\\\\ \\\\
r= \sqrt{(- \frac{5}{2})^{2} +0^{2}+ \frac{11}{4}} \\\\\\\
r= \sqrt{\frac{25}{4} +0+ \frac{11}{4}} } \\\\\
r= \sqrt{ \frac{25+11}{4} } \\\\\\\\
r= \sqrt{ \frac{36}{4} }\\\\\
r= \sqrt{9} \\\\\
r=3\\\\\\\\\
C( \frac{5}{2},0)\\\\\
r=3](https://tex.z-dn.net/?f=A%29%5C%5C%5C%5C%5C%0A+x%5E%7B2%7D+-5x%2B+y%5E%7B2%7D-+%5Cfrac%7B11%7D%7B4%7D%3D0%5C%5C%5C%5C%5C%0A+x%5E%7B2%7D+-5x%2B+y%5E%7B2%7D%3D+%5Cfrac%7B11%7D%7B4%7D%5C%5C%5C%5C%5C+%0A%28x-+%5Cfrac%7B5%7D%7B2%7D%29%5E%7B2%7D%2B%28y%2B0%29%5E%7B2%7D%3D+%5Cfrac%7B11%7D%7B4%7D%5C%5C%5C%5C%5C+%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B%28-+%5Cfrac%7B5%7D%7B2%7D%29%5E%7B2%7D+%2B0%5E%7B2%7D%2B+%5Cfrac%7B11%7D%7B4%7D%7D+%5C%5C%5C%5C%5C%5C%5C%0A+r%3D+%5Csqrt%7B%5Cfrac%7B25%7D%7B4%7D+%2B0%2B+%5Cfrac%7B11%7D%7B4%7D%7D+%7D+%5C%5C%5C%5C%5C%0Ar%3D++%5Csqrt%7B+%5Cfrac%7B25%2B11%7D%7B4%7D+%7D+%5C%5C%5C%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B+%5Cfrac%7B36%7D%7B4%7D+%7D%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B9%7D+%5C%5C%5C%5C%5C%0Ar%3D3%5C%5C%5C%5C%5C%5C%5C%5C%5C%0AC%28+%5Cfrac%7B5%7D%7B2%7D%2C0%29%5C%5C%5C%5C%5C%0Ar%3D3++ "A)\\\\\
x^{2} -5x+ y^{2}- \frac{11}{4}=0\\\\\
x^{2} -5x+ y^{2}= \frac{11}{4}\\\\\
(x- \frac{5}{2})^{2}+(y+0)^{2}= \frac{11}{4}\\\\\ \\\\
r= \sqrt{(- \frac{5}{2})^{2} +0^{2}+ \frac{11}{4}} \\\\\\\
r= \sqrt{\frac{25}{4} +0+ \frac{11}{4}} } \\\\\
r= \sqrt{ \frac{25+11}{4} } \\\\\\\\
r= \sqrt{ \frac{36}{4} }\\\\\
r= \sqrt{9} \\\\\
r=3\\\\\\\\\
C( \frac{5}{2},0)\\\\\
r=3")
![B)\\\\
x^{2} +y ^{2}+x-6y- \frac{15}{4}=0\\\\
x^{2} +x+y^{2}-6y= \frac{15}{4} \\\\\
(x+ \frac{1}{2})^{2}+(y-3)^{2}=\frac{15}{4} \\\\\\\\\\
r= \sqrt{ (\frac{1}{2})^{2}+(-3)^{2}+ \frac{15}{4}}\\\\\
r= \sqrt{ \frac{1}{4}+9+ \frac{15}{4}}\\\\\
r= \sqrt{\frac{1+36+15}{4}}\\\\\
r= \sqrt{ \frac{52}{4}}\\\\\
r= \sqrt{13}\\\\\\\\
C( -\dfrac{1}{2},3)\\\\
r= \sqrt{13}](https://tex.z-dn.net/?f=B%29%5C%5C%5C%5C%0A+x%5E%7B2%7D+%2By+%5E%7B2%7D%2Bx-6y-+%5Cfrac%7B15%7D%7B4%7D%3D0%5C%5C%5C%5C%0A+x%5E%7B2%7D+%2Bx%2By%5E%7B2%7D-6y%3D+%5Cfrac%7B15%7D%7B4%7D+%5C%5C%5C%5C%5C%0A%28x%2B+%5Cfrac%7B1%7D%7B2%7D%29%5E%7B2%7D%2B%28y-3%29%5E%7B2%7D%3D%5Cfrac%7B15%7D%7B4%7D+%5C%5C%5C%5C%5C%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B+%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7B2%7D%2B%28-3%29%5E%7B2%7D%2B+%5Cfrac%7B15%7D%7B4%7D%7D%5C%5C%5C%5C%5C+++%0A+r%3D+%5Csqrt%7B+%5Cfrac%7B1%7D%7B4%7D%2B9%2B+%5Cfrac%7B15%7D%7B4%7D%7D%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B%5Cfrac%7B1%2B36%2B15%7D%7B4%7D%7D%5C%5C%5C%5C%5C%0Ar%3D++%5Csqrt%7B+%5Cfrac%7B52%7D%7B4%7D%7D%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B13%7D%5C%5C%5C%5C%5C%5C%5C%5C%0A%0AC%28+-%5Cdfrac%7B1%7D%7B2%7D%2C3%29%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B13%7D++++%0A%0A++++ "B)\\\\
x^{2} +y ^{2}+x-6y- \frac{15}{4}=0\\\\
x^{2} +x+y^{2}-6y= \frac{15}{4} \\\\\
(x+ \frac{1}{2})^{2}+(y-3)^{2}=\frac{15}{4} \\\\\\\\\\
r= \sqrt{ (\frac{1}{2})^{2}+(-3)^{2}+ \frac{15}{4}}\\\\\
r= \sqrt{ \frac{1}{4}+9+ \frac{15}{4}}\\\\\
r= \sqrt{\frac{1+36+15}{4}}\\\\\
r= \sqrt{ \frac{52}{4}}\\\\\
r= \sqrt{13}\\\\\\\\
C( -\dfrac{1}{2},3)\\\\
r= \sqrt{13}")
Exercício 2
![9 x^{2} +9y^{2}+6x-36y+64=0\\\\
\frac{9 x^{2} }{9}+ \frac{9y^{2} }{9}+ \frac{6x}{9}- \frac{36y}{9} + \frac{64}{9}=0\\\\\
x^{2} +y^{2}+ \frac{2x}{3}-4y=- \frac{64}{9}\\\\\
(x+ \frac{1}{3})^{2} +(y-2)^{2}= -\frac{64}{9}\\\\\\\\
r= \sqrt{ ( \frac{1}{3})^{2}+(-2)^{2}- \frac{64}{9}}\\\\\\
r= \sqrt{ \frac{1}{9}+4- \frac{64}{9}}\\\\\\
r= \sqrt{ \frac{1+36-64}{4}} \\\\\
r= \sqrt{ -\frac{27}{4} } \\\\\\\](https://tex.z-dn.net/?f=9+x%5E%7B2%7D+%2B9y%5E%7B2%7D%2B6x-36y%2B64%3D0%5C%5C%5C%5C%0A+%5Cfrac%7B9+x%5E%7B2%7D+%7D%7B9%7D%2B+%5Cfrac%7B9y%5E%7B2%7D+%7D%7B9%7D%2B+%5Cfrac%7B6x%7D%7B9%7D-+%5Cfrac%7B36y%7D%7B9%7D+%2B+%5Cfrac%7B64%7D%7B9%7D%3D0%5C%5C%5C%5C%5C%0A+x%5E%7B2%7D+%2By%5E%7B2%7D%2B+%5Cfrac%7B2x%7D%7B3%7D-4y%3D-+%5Cfrac%7B64%7D%7B9%7D%5C%5C%5C%5C%5C%0A%28x%2B+%5Cfrac%7B1%7D%7B3%7D%29%5E%7B2%7D+%2B%28y-2%29%5E%7B2%7D%3D+-%5Cfrac%7B64%7D%7B9%7D%5C%5C%5C%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B+%28+%5Cfrac%7B1%7D%7B3%7D%29%5E%7B2%7D%2B%28-2%29%5E%7B2%7D-+%5Cfrac%7B64%7D%7B9%7D%7D%5C%5C%5C%5C%5C%5C+++%0Ar%3D+%5Csqrt%7B+%5Cfrac%7B1%7D%7B9%7D%2B4-+%5Cfrac%7B64%7D%7B9%7D%7D%5C%5C%5C%5C%5C%5C%0Ar%3D++%5Csqrt%7B+%5Cfrac%7B1%2B36-64%7D%7B4%7D%7D+%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B+-%5Cfrac%7B27%7D%7B4%7D+%7D+%5C%5C%5C%5C%5C%5C%5C%0A%0A%0A%0A%0A+++++++++ "9 x^{2} +9y^{2}+6x-36y+64=0\\\\
\frac{9 x^{2} }{9}+ \frac{9y^{2} }{9}+ \frac{6x}{9}- \frac{36y}{9} + \frac{64}{9}=0\\\\\
x^{2} +y^{2}+ \frac{2x}{3}-4y=- \frac{64}{9}\\\\\
(x+ \frac{1}{3})^{2} +(y-2)^{2}= -\frac{64}{9}\\\\\\\\
r= \sqrt{ ( \frac{1}{3})^{2}+(-2)^{2}- \frac{64}{9}}\\\\\\
r= \sqrt{ \frac{1}{9}+4- \frac{64}{9}}\\\\\\
r= \sqrt{ \frac{1+36-64}{4}} \\\\\
r= \sqrt{ -\frac{27}{4} } \\\\\\\")
Raio negativo tal situação não existe dentro da matemática,logo concluímos que a equação não é de uma circunferência.
![x^{2} +y^{2}-3y= \frac{7}{4} \\\\\\
(x+0)^{2}+(y -\frac{3}{2})^{2}= \frac{7}{4}\\\\\\
r= \sqrt{0^{2}+( -\frac{3}{2}) ^{2}+ \frac{7}{4} }\\\\\\\
r= \sqrt{\frac{9}{4} + \frac{7}{4} }\\\\\\\
r= \sqrt{ \frac{16}{4} } \\\\\
r= \sqrt{4}\\\\
r=2\\\\\\
C(0, \frac{3}{2})\\\\\
r=2](https://tex.z-dn.net/?f=+x%5E%7B2%7D+%2By%5E%7B2%7D-3y%3D+%5Cfrac%7B7%7D%7B4%7D+%5C%5C%5C%5C%5C%5C%0A%28x%2B0%29%5E%7B2%7D%2B%28y+-%5Cfrac%7B3%7D%7B2%7D%29%5E%7B2%7D%3D+%5Cfrac%7B7%7D%7B4%7D%5C%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B0%5E%7B2%7D%2B%28+-%5Cfrac%7B3%7D%7B2%7D%29+%5E%7B2%7D%2B+%5Cfrac%7B7%7D%7B4%7D+%7D%5C%5C%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B%5Cfrac%7B9%7D%7B4%7D+%2B+%5Cfrac%7B7%7D%7B4%7D+%7D%5C%5C%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B++%5Cfrac%7B16%7D%7B4%7D+%7D+%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B4%7D%5C%5C%5C%5C%0Ar%3D2%5C%5C%5C%5C%5C%5C%0AC%280%2C+%5Cfrac%7B3%7D%7B2%7D%29%5C%5C%5C%5C%5C%0Ar%3D2++%0A++++++++++++ "x^{2} +y^{2}-3y= \frac{7}{4} \\\\\\
(x+0)^{2}+(y -\frac{3}{2})^{2}= \frac{7}{4}\\\\\\
r= \sqrt{0^{2}+( -\frac{3}{2}) ^{2}+ \frac{7}{4} }\\\\\\\
r= \sqrt{\frac{9}{4} + \frac{7}{4} }\\\\\\\
r= \sqrt{ \frac{16}{4} } \\\\\
r= \sqrt{4}\\\\
r=2\\\\\\
C(0, \frac{3}{2})\\\\\
r=2")
É uma circunferência.
Exercício 3
Retas tangente a circunferência.
![m=2\\\\
P_{1}(-2,1)\\\\\
y-1=2(x-2)\\\\\
y-1=2x+4\\\\\
y=2x+4+1\\\\\
y=2x+5\\\\\\\\
m=2\\\\\
P_{2}(2,-1)\\\\\
y+1=2(x-2)\\\\\
y+1=2x-4\\\\\
y=2x-4-1\\\\
y=2x-5](https://tex.z-dn.net/?f=m%3D2%5C%5C%5C%5C%0AP_%7B1%7D%28-2%2C1%29%5C%5C%5C%5C%5C%0Ay-1%3D2%28x-2%29%5C%5C%5C%5C%5C%0Ay-1%3D2x%2B4%5C%5C%5C%5C%5C%0Ay%3D2x%2B4%2B1%5C%5C%5C%5C%5C%0Ay%3D2x%2B5%5C%5C%5C%5C%5C%5C%5C%5C%0A%0Am%3D2%5C%5C%5C%5C%5C%0AP_%7B2%7D%282%2C-1%29%5C%5C%5C%5C%5C%0Ay%2B1%3D2%28x-2%29%5C%5C%5C%5C%5C%0Ay%2B1%3D2x-4%5C%5C%5C%5C%5C%0Ay%3D2x-4-1%5C%5C%5C%5C%0Ay%3D2x-5+%0A+ "m=2\\\\
P_{1}(-2,1)\\\\\
y-1=2(x-2)\\\\\
y-1=2x+4\\\\\
y=2x+4+1\\\\\
y=2x+5\\\\\\\\
m=2\\\\\
P_{2}(2,-1)\\\\\
y+1=2(x-2)\\\\\
y+1=2x-4\\\\\
y=2x-4-1\\\\
y=2x-5")
Boa tarde!
Bons estudos
Solução!
Para encontrar o centro de uma circunferencia basta comparar.
Exercício 2
Raio negativo tal situação não existe dentro da matemática,logo concluímos que a equação não é de uma circunferência.
É uma circunferência.
Exercício 3
Retas tangente a circunferência.
Boa tarde!
Bons estudos
Anexos:
DaphinneMartins:
Obrigado msm !
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