A circle has a center that falls on the line $y = \frac{2}{3} x + 7$ $y = 2/3x +7$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(6, 4)$ $(6 ,4 )$ . What is the equation of the circle?

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A circle has a center that falls on the line $y = \frac{2}{3} x + 7$ $y = 2/3x +7$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(6, 4)$ $(6 ,4 )$ . What is the equation of the circle?

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Answer 1 · 2016-05-14T12:17:07

The equation of the circle is $x^{2} + y^{2} - 14 y + 4 = 0$ $x^2+y^2-14y+4=0$

Explanation:

As the center will be equidistant from $(3, 1)$ $(3,1)$ and $(6, 4)$ $(6,4)$ , the center will lie on the line

${(x - 3)}^{2} + {(y - 1)}^{2} = {(x - 6)}^{2} + {(y - 4)}^{2}$ $(x-3)^2+(y-1)^2=(x-6)^2+(y-4)^2$ or

$x^{2} - 6 x + 9 + y^{2} - 2 y + 1 = x^{2} - 12 x + 36 + y^{2} - 8 y + 16$ $x^2-6x+9+y^2-2y+1=x^2-12x+36+y^2-8y+16$

$12 x - 6 x + 8 y - 2 y + 10 - 52 = 0$ $12x-6x+8y-2y+10-52=0$ or $6 x + 6 y - 42 = 0$ $6x+6y-42=0$ or

$x + y = 7$ $x+y=7$ but as it also lies on $y = \frac{2}{3} x + 7$ $y=2/3x+7$ , putting this in first

$x + \frac{2}{3} x + 7 = 7$ $x+2/3x+7=7$ or $x = 0$ $x=0$ and hence $y = 7$ $y=7$ . Hence center is $(0, 7)$ $(0,7)$ .

Its distance from $(3, 1)$ $(3,1)$ is $\sqrt{{(0 - 3)}^{2} + {(7 - 1)}^{2}} = \sqrt{9 + 36} = \sqrt{45}$ $sqrt((0-3)^2+(7-1)^2)=sqrt(9+36)=sqrt45$ , i.e radius

Hence equation of circle is

${(x - 0)}^{2} + {(y - 7)}^{2} = {(\sqrt{45})}^{2}$ $(x-0)^2+(y-7)^2=(sqrt45)^2$

$x^{2} + y^{2} - 14 y + 49 = 45$ $x^2+y^2-14y+49=45$ or $x^{2} + y^{2} - 14 y + 4 = 0$ $x^2+y^2-14y+4=0$

graph{(x^2+y^2-14y+4)(y-(2/3)x-7)(x+y-7)=0 [-17.5, 22.5, -4.24, 15.76]}

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A circle has a center that falls on the line $y = \frac{2}{3} x + 7$ $y = 2/3x +7$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(6, 4)$ $(6 ,4 )$ . 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=23x+7y = 2/3x +7 and passes through (3,1) ( 3 ,1 ) and (6,4)(6 ,4 ). What is the equation of the circle?

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A circle has a center that falls on the line $y = \frac{2}{3} x + 7$ $y = 2/3x +7$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(6, 4)$ $(6 ,4 )$ . What is the equation of the circle?