Question

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1) Determine a equação da circunferência de centro e raio dados e faça os gráficos.

a) centro (3,7/2), raio 10/3.

b) centro ( -17/3, 13/6), raio $\sqrt{ \frac{7}{2} }$


2) determine as interseções da reta x - $\sqrt{3y} + 4=0$ com a circunferência x² + y² =16 e faça o gráfico.

Answer 1

Boa tarde!


Solução!


Equação da circunferência na forma algébrica reduzida.


$(x-a)^{2}+(y-b)^{2}=r^{2}$


$a) ~~centro (3, \frac{7}{2} ), raio \frac{10}{3} .\\\\\\ (x-3)^{2} +( y-\frac{7}{3})^{2}= (\frac{10}{3})^{2}\\\\\ \\ (x-3)^{2} +( y-\frac{7}{3})^{2}= \frac{100}{9}$



$b) centro ( -\frac{17}{3} , \frac{13}{6} ), raio \sqrt{ \frac{7}{2} }\\\\\\\\\ (x+ \frac{17}{3})^{2}+(y- \frac{13}{6})^{2} = \sqrt{ \frac{7}{2} }\\\\\\\ (x+ \frac{17}{3})^{2}+(y- \frac{13}{6})^{2} = (\sqrt{ \frac{7}{2} })^{2} \\\\\\\ (x+ \frac{17}{3})^{2}+(y- \frac{13}{6})^{2} = \frac{7}{2}$




2) determine as interseções da reta$x- \sqrt{3}y+4=0$  com a circunferência x² + y² =16 e faça o gráfico.


$x- \sqrt{3}y+4=0\\\\\ x+4= \sqrt{3}y\\\\\\ \dfrac{x+4}{ \sqrt{3}}=y\\\\\\\ x^{2} + y^{2}=16\\\\ Substituindo\\\\\ x^{2} +(\dfrac{x+4}{ \sqrt{3}})^{2}=16\\\\\\\\ x^{2} + \dfrac{ x^{2} +8x+16}{3}=16\\\\\\\\\ 3 x^{2} + x^{2} +8x+16=48\\\\\ 3 x^{2} + x^{2} +8x+16-48=0\\\\\ 3 x^{2} + x^{2} +8x-32=0\\\\\ 4 x^{2} +8x-32=0\\\\\ Vamos~~dividir~~a~~equacao~~por~~4\\\\\ 4 x^{2} +8x-32=0:(4)\\\\\ x^{2} +2x-8=0$


Vamos usar a formula de Bhaskara!

$x= \dfrac{-2\pm \sqrt{2^{2}-4.1.(-8) } }{2.1}\\\\\ x= \dfrac{-2\pm \sqrt{4+32 } }{2}\\\\\ x= \dfrac{-2\pm \sqrt{36 } }{2}\\\\\ x= \dfrac{-2\pm 6 }{2}\\\\\\\ x_{1}= \dfrac{-2+6}{2}= \dfrac{4}{2}=2\\\\\\\ x_{2} = \dfrac{-4-6}{2y}= \dfrac{-8}{2} =-4$


Pontos de intersecção da reta com a circunferência.

$x^{2} +y^{2}=16~~~~~~~~ x_{1} =2\\\\\\\\ y^{2} =16- x^{2}\\\\\ y^{2} =16- (2)^{2}\\\\\ y^{2} =16- 4\\\\\ y^{2} =12\\\\\ y= \sqrt{12}\\\\\ y=2 \sqrt{3} \\\\\ P_{1}(2,2 \sqrt{3})$


$x^{2} +y^{2}=16~~~~~~~~ x_{2} =-4\\\\\ y^{2} =16- (-4)^{2}\\\\\ y^{2} =16- 16\\\\ y= \sqrt{16-16}\\\\\ y= \sqrt{0}\\\\\ y=0\\\\\\ P_{2} (-4,0)$



Boa noite!

Bons estudos!