A circle has a center that falls on the line $y = \frac{8}{9} x + 4$ $y = 8/9x +4$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(5, 7)$ $(5 ,7 )$ . What is the equation of the circle?

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A circle has a center that falls on the line $y = \frac{8}{9} x + 4$ $y = 8/9x +4$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(5, 7)$ $(5 ,7 )$ . What is the equation of the circle?

1 Answer

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Answer 1 · 2016-10-26T11:05:15

The equation of the circle is
${(x - 1.091)}^{2} + {(y - 4.97)}^{2} = {4.41}^{2}$ $(x-1.091)^2+(y-4.97)^2=4.41^2$

Explanation:

Let the center of the circle be $(a, b)$ $(a,b)$ and the radius $r$ $r$
The the equation of the circle is ${(x - a)}^{2} + {(y - b)}^{2} = r^{2}$ $(x-a)^2+(y-b)^2=r^2$

The circle passes through the two points. So
${(3 - a)}^{2} + {(1 - b)}^{2} = r^{2}$ $(3-a)^2+(1-b)^2=r^2$
and
${(5 - a)}^{2} + {(7 - b)}^{2} = r^{2}$ $(5-a)^2+(7-b)^2=r^2$

We can equalize the two equations

${(3 - a)}^{2} + {(1 - b)}^{2} = {(5 - a)}^{2} + {(7 - b)}^{2}$ $(3-a)^2+(1-b)^2=(5-a)^2+(7-b)^2$
Developing and simplifying
$9 - 6 a + a^{2} + 1 - 2 b + b^{2} = 25 - 10 a + a^{2} + 49 - 14 b + b^{2}$ $9-6a+a^2+1-2b+b^2=25-10a+a^2+49-14b+b^2$

$10 - 6 a - 2 b = 74 - 10 a - 14 b$ $10-6a-2b=74-10a-14b$

$12 b = - 4 a + 64$ $12b=-4a+64$ $\Rightarrow$ $=>$ $3 b = - a + 16$ $3b=-a+16$ , this equation 1

We obtain the equation 2 from the equation of the line
$b = \frac{8 a}{9} + 4$ $b=(8a)/9+4$

Solving for $a$ $a$ and $b$ $b$ , we get $a = 1.091$ $a=1.091$ and $b = 4.97$ $b=4.97$
and the radius of the circle is

$r^{2} = {(3 - 1.091)}^{2} + {(1 - 4.97)}^{2}$ $r^2=(3-1.091)^2+(1-4.97)^2$
$r^{2} = {1.909}^{2} + {3.97}^{2}$ $r^2=1.909^2+3.97^2$

$r = 4.41$ $r=4.41$
So the equation of the circle is
${(x - 1.091)}^{2} + {(y - 4.97)}^{2} = {4.41}^{2}$ $(x-1.091)^2+(y-4.97)^2=4.41^2$

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A circle has a center that falls on the line $y = \frac{8}{9} x + 4$ $y = 8/9x +4$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(5, 7)$ $(5 ,7 )$ . What is the equation of the circle?

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Explanation:

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A circle has a center that falls on the line y = 8/9x +4 y=89x+4y = 8/9x +4 and passes through ( 3 ,1 )(3,1) ( 3 ,1 ) and (5 ,7 )(5,7)(5 ,7 ). What is the equation of the circle?

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Explanation:

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A circle has a center that falls on the line $y = \frac{8}{9} x + 4$ $y = 8/9x +4$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(5, 7)$ $(5 ,7 )$ . What is the equation of the circle?