Question 1046324
B is some point, (x,y).  From there, use the DISTANCE FORMULA, and the equation of a LINE containing two given points.


{{{AB=2BC}}}
{{{sqrt((x-0)^2+(y-(-3))^2)=2*sqrt((x-3)^2+(y-3)^2)}}}
{{{sqrt(x^2+(y+3)^2)=2*sqrt((x-3)^2+(y-3)^2)}}}

{{{x^2+(y+3)^2=4((x-3)^2+(y-3)^2)}}}

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Maybe using the equation of the line would be more convenient before carrying all those multiplications from that previous equation.


{{{y=mx-3}}}
{{{m=(3-(-3))/(3-0)}}}
{{{m=6/3}}}
{{{m=2}}}
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{{{y=2x-3}}}--------Substitute for y in the equation which came from Distance Formula result.


{{{x^2+(2x-3+3)^2=4((x-3)^2+(2x-3-3)^2)}}}
{{{x^2+(2x)^2=4((x-3)^2+(2x-6)^2)}}}
{{{5x^2=4((x-3)^2+2^2(x-3)^2)}}}-----see carefully how there is a factor of x-3 distributed in the right member.
{{{5x^2=4(x-3)^2(1+4)}}}
{{{5x^2=4*5*(x-3)^2}}}------common factor 5 on both sides
{{{x^2=4(x-3)^2}}}
{{{x^2=4(x^2-6x+9)}}}
{{{x^2=4x^2-24x+36}}}
{{{3x^2-24x+36=0}}}
Divide both sides by 3.
{{{x^2-8x+12=0}}}
{{{highlight((x-6)(x-2)=0)}}}
{{{highlight_green(system(x=6,or,x=2))}}}------------------------------evaluate the corresponding y values.  Use the linear equation for this.  Choose the combination, or point which is BETWEEN points A and C.



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How to know that the line containing points A and C has y-intercept of -3?
The problem description clearly says so.  This was given as point A.