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elena-14-01-66 [18.8K]
1 year ago
6

Find the center and radius of the circle

Mathematics
2 answers:
oksian1 [2.3K]1 year ago
6 0
<h2>Centre: </h2>

  • The general formula for the circumference is:

\boxed{ \boxed{{x^{2} \:  +  \: y ^{2}  \: +  \: Dx  \: + \:  Ey  \: + F \:  =  \: 0}}}

________________________

To find the center, write this formula:

\boxed{ \boxed{C( \frac{-D}{2} ,  \frac{-E}{2} )}}

____________________

<h3>We know that...</h3>

D \: = \: -6  \\  E \: = \: 4  \\  F \: = \: 4

We use the equation of the center with the values already obtained:

C( \frac{- - 6}{2} ,  \frac{-4}{2} )

C( \frac{6}{2} ,  \frac{-4}{2} )

\huge\boxed{ \bold{C( 3,   - 2 )}}

_____________________

<h2>Ratio: </h2>

Now we use the equation of the radius for a circumference, which is:

\boxed{ \boxed{{r \:  =  \:  \frac{1}{2}  \sqrt{D ^{2}  \:  +  \: E ^{2} \:  -  \: 4F } }}}

___________________________

Now we use the equation of the radius for a circumference with the values already obtained.

r \:  =  \:  \frac{1}{2}  \sqrt{  { - 6}^{2}  \:  +  \:  {4}^{2}  \:  -  \: 4 \:  \times  \: 4 }

  • I am going to use complex numbers because the square root of a negative number does not exist in the set of real numbers.

r \:  =  \:  \frac{1}{2}  \sqrt{  { - 6}^{2}  \:   -   \:  {4}^{2}  \:  - +  \: 4 \:  \times  \: 4 i}

r \:  =  \:  \frac{1}{2}  \sqrt{  {  6}^{2}  \:   -   \:  {4}^{2}  \:   +  \: 16i}

r \:  =  \:  \frac{ \sqrt{ {6}^{2} \:  -  \:  {4}^{2}   \:  +  \: 16i} }{2}

r \:  =  \:  \frac{ \sqrt{36 \:  -  \: 16 \:  +  \: 16i} }{2}

r \:  =  \:   \frac{ \sqrt{36i} }{2}

r \:  =  \:  \frac{6i}{2}

\huge \boxed{ \bold{r \:  =  \: 3i}}

<h3><em><u>MissSpanish</u></em></h3>

Wewaii [24]1 year ago
3 0
<h2><u>AnswEr </u><u>:</u></h2>

Provided Equation

  • x² + y² - 6x + 4y + 4= 0

As we know the Standard Equation of a Circle is

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

where,

r radius of circle.

(a,b) centre .

\implies \sf \:  {x}^{2}  +  {y}^{2}  - 6x + 4y + 4 = 0

This equation can be further written as

\implies \sf \:  {x}^{2}   - 6x +   {y}^{2}   + 4y  =  - 4

Now completing the square ( by adding 4 & 9 on both side ) .

\implies \sf \:  {x}^{2}   - 6x +  9 +  {y}^{2}   + 4y  + 4  =  - 4 + 9  + 4

again this equation can be further written as

\implies \sf \: (x - 3) {}^{2}  + (y + 2) {}^{2}  =  {3}^{2}

Now comparing this equation with standard equation of circle ( mentioned above) and we will get

  • Centre = ( a,b) = (3, -2 )
  • Radius = r = 3

Therefore,

  • <u>Centre </u><u>of </u><u>circle </u><u>is </u><u>(</u><u>3</u><u>,</u><u>-</u><u>2</u><u>)</u><u> </u><u>and </u><u>radius </u><u>is </u><u>3</u>
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3- The solution of I5 - xI = 6 is {-1 , 11}

4- The solution of I6x - 8I + 7 = 5 is ∅

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* Now lets find the solution of each equation

1- ∵ I2x + 5I = 9

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∴ 2x + 5 = -9 ⇒ subtract 5 from both sides

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* The solution of I2x + 5I = 9 is {-7 , 2}

2- ∵ I2x + 7I + 2 = 11 ⇒ Subtract 2 from both sides

∴ I2x + 7I = 9

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∴ 2x = 2 ⇒ divide both sides by 2

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* The solution of Ix + 3I = 12 is {-15 , 9}

6- ∵ Ix - 3I = -12

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