A circle has a center that falls on the line $y = 7/6x +1$ and passes through $(1 ,4 )$ and $(8 ,5 )$ . What is the equation of the circle?

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Answer 1 · 2017-06-19T13:55:30

The equation of the circle is $(x-210/49)^2+(y-294/49)^2=35525/2401$

Let $C$ be the mid point of $A=(1,4)$ and $B=(8,5)$

$C=((1+8)/2,(4+5)/2)=(9/2,9/2)$

The slope of $AB$ is $=(5-4)/(8-1)=1/7$

The slope of the line perpendicular to $AB$ is $=-7$

The equation of the line passing trrough $C$ and perpendicular to $AB$ is

$y-9/2=-7(x-9/2)$

$y=-7x+63/2+9/2=-7x+36$

The intersection of this line with the line $y=7/6x+1$ gives the center of the circle.

$7/6x+1=-7x+36$

$7/6x+7x=36-1$

$49/6x=35$

$x=6*35/49=210/49$

$y=-7*210/49+36=294/49$

The center of the circle is $(210/49,294/49)$

The radius of the circle is

$r^2=(1-210/49)^2+(4-294/49)^2$

$=(161/49)^2+(98/49)^2$

$=35525/2401$

The equation of the circle is

$(x-210/49)^2+(y-294/49)^2=35525/2401$
graph{((x-210/49)^2+(y-294/49)^2-35525/2401)(y-7/6x-1)=0 [-11.24, 14.07, -1.11, 11.55]}

A circle has a center that falls on the line y = 7/6x +1 y = 7/6x +1 and passes through (1 ,4 )(1 ,4 ) and (8 ,5 )(8 ,5 ). What is the equation of the circle?