A circle has a center that falls on the line $y = \frac{7}{2} x + 3$ $y = 7/2x +3$ and passes through $(1, 2)$ $(1 ,2 )$ and $(8, 5)$ $(8 ,5 )$ . What is the equation of the circle?

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Answer 1 · 2017-07-07T09:52:15

The equation of the circle is ${(x - \frac{66}{35})}^{2} + {(y - \frac{48}{5})}^{2} = \frac{71717}{1225}$ $(x-66/35)^2+(y-48/5)^2=71717/1225$

Let $C$ $C$ be the mid point of $A = (1, 2)$ $A=(1,2)$ and $B = (8, 5)$ $B=(8,5)$

$C = (\frac{1 + 8}{2}, \frac{5 + 2}{2}) = (\frac{9}{2}, \frac{7}{2})$ $C=((1+8)/2,(5+2)/2)=(9/2,7/2)$

The slope of $A B$ $AB$ is $= \frac{5 - 2}{8 - 1} = \frac{3}{7}$ $=(5-2)/(8-1)=(3)/(7)$

The slope of the line perpendicular to $A B$ $AB$ is $= - \frac{7}{3}$ $=-7/3$

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

$y - \frac{7}{2} = - \frac{7}{3} (x - \frac{9}{2})$ $y-7/2=-7/3(x-9/2)$

$y = - \frac{7}{3} x + \frac{21}{2} + \frac{7}{2} = - \frac{7}{3} x + 14$ $y=-7/3x+21/2+7/2=-7/3x+14$

The intersection of this line with the line $y = \frac{7}{2} x + 3$ $y=7/2x+3$ gives the center of the circle.

$- \frac{7}{3} x + 14 = \frac{7}{2} x + 3$ $-7/3x+14=7/2x+3$

$\frac{7}{2} x + \frac{7}{3} x = 14 - 3$ $7/2x+7/3x=14-3$

$\frac{35}{6} x = 11$ $35/6x=11$

$x = \frac{66}{35}$ $x=66/35$

$y = \frac{7}{2} \cdot (\frac{66}{35}) + 3 = \frac{48}{5}$ $y=7/2*(66/35)+3=48/5$

The center of the circle is $(\frac{66}{35}, \frac{48}{5})$ $(66/35,48/5)$

The radius of the circle is

$r^{2} = {(1 - \frac{66}{35})}^{2} + {(2 - \frac{48}{5})}^{2}$ $r^2=(1-66/35)^2+(2-48/5)^2$

$= {(- \frac{31}{35})}^{2} + {(- \frac{38}{5})}^{2}$ $=(-31/35)^2+(-38/5)^2$

$= \frac{11069}{1225}$ $=11069/1225$

The equation of the circle is

${(x - \frac{66}{35})}^{2} + {(y - \frac{48}{5})}^{2} = \frac{71717}{1225}$ $(x-66/35)^2+(y-48/5)^2=71717/1225$

graph{((x-66/35)^2+(y-48/5)^2-71717/1225)(y-7/2x-3)(y+7/3x-14)=0 [-23.05, 12.98, -0.97, 17.06]}

A circle has a center that falls on the line y = 7/2x +3 y=72x+3y = 7/2x +3 and passes through (1 ,2 )(1,2)(1 ,2 ) and (8 ,5 )(8,5)(8 ,5 ). What is the equation of the circle?