Question

Em cada item abaixo, determine as equações geral e reduzida das circunferências :
a) de centro (0, 0) e raio 6.
b) de centro (2, 2) e raio 3.
c) de centro (−1, -8) e raio 5.

Answer 1

Equação reduzida da circunferência

$\boxed{\boxed{\mathsf{{(x-a)}^{2}+{(y-b)}^{2}={R}^{2}}}}$

Equação geral da

circunferência

${x}^{2}+{y}^{2}-2ax-2by+{a}^{2}+{b}^{2}-{R}^{2} \\ =0$

a) C(a, b) =(0,0)

a=0

b=0

R=6

$\mathsf{{(x-0)}^{2}+{(y-0)}^{2}={6}^{2}}$

$\boxed{\boxed{\mathsf{{(x)}^{2}+{(y)}^{2}=36}}}$

${x}^{2}+{y}^{2}-2.0.x-2.0.y+{0}^{2}+{0}^{2}-{6}^{2}\\=0$

${x}^{2}+{y}^{2}-36=0$

b)

a=2

b=2

R=3

$\mathsf{{(x-2)}^{2}+{(y-2)}^{2}={3}^{2}}$

$\boxed{\boxed{\mathsf{{(x-2)}^{2}+{(y-2)}^{2}=9}}}$

${x}^{2}+{y}^{2}-2.2.x-2.2.y+{2}^{2}+{2}^{2}-{3}^{2}\\=0$

${x}^{2}+{y}^{2}-4x-4y-1=0$

c)

a=-1

b=-8

R=5

$\mathsf{{(x-[-1])}^{2}+{(y-[-8])}^{2}={5}^{2}}$

$\boxed{\boxed{\mathsf{{(x+1)}^{2}+{(y+8)}^{2}=25}}}$

${x}^{2}+{y}^{2}-2.(-1).x-2.(-8).y={(-2)}^{2}+{(-8)}^{2}-{5}^{2}=0v$

${x}^{2}+{y}^{2}+2x+16y+43=0$

Answer 2

Resposta:

Equação reduzida da circunferência

\boxed{\boxed{\mathsf{{(x-a)}^{2}+{(y-b)}^{2}={R}^{2}}}}(x−a)2+(y−b)2=R2

Equação geral da

circunferência

\begin{lgathered}{x}^{2}+{y}^{2}-2ax-2by+{a}^{2}+{b}^{2}-{R}^{2} \\ =0\end{lgathered}x2+y2−2ax−2by+a2+b2−R2=0

a) C(a, b) =(0,0)

a=0

b=0

R=6

\mathsf{{(x-0)}^{2}+{(y-0)}^{2}={6}^{2}}(x−0)2+(y−0)2=62

\boxed{\boxed{\mathsf{{(x)}^{2}+{(y)}^{2}=36}}}(x)2+(y)2=36

\begin{lgathered}{x}^{2}+{y}^{2}-2.0.x-2.0.y+{0}^{2}+{0}^{2}-{6}^{2}\\=0\end{lgathered}x2+y2−2.0.x−2.0.y+02+02−62=0

{x}^{2}+{y}^{2}-36=0x2+y2−36=0

b)

a=2

b=2

R=3

\mathsf{{(x-2)}^{2}+{(y-2)}^{2}={3}^{2}}(x−2)2+(y−2)2=32

\boxed{\boxed{\mathsf{{(x-2)}^{2}+{(y-2)}^{2}=9}}}(x−2)2+(y−2)2=9

\begin{lgathered}{x}^{2}+{y}^{2}-2.2.x-2.2.y+{2}^{2}+{2}^{2}-{3}^{2}\\=0\end{lgathered}x2+y2−2.2.x−2.2.y+22+22−32=0

{x}^{2}+{y}^{2}-4x-4y-1=0x2+y2−4x−4y−1=0

c)

a=-1

b=-8

R=5

\mathsf{{(x-[-1])}^{2}+{(y-[-8])}^{2}={5}^{2}}(x−[−1])2+(y−[−8])2=52

\boxed{\boxed{\mathsf{{(x+1)}^{2}+{(y+8)}^{2}=25}}}(x+1)2+(y+8)2=25

{x}^{2}+{y}^{2}-2.(-1).x-2.(-8).y={(-2)}^{2}+{(-8)}^{2}-{5}^{2}=0vx2+y2−2.(−1).x−2.(−8).y=(−2)2+(−8)2−52=0v

{x}^{2}+{y}^{2}+2x+16y+43=0x2+y2+2x+16y+43=0