Question 774946
[ a point is at a distance {{{4sqrt(2)}}} units from (-3/2,-5/2) and at a distance {{{2*sqrt(5)}}} units from (9/2,-5/2), find the point ]


This point is unknown (x,y).


Look for one or two triangles with one side {{{9/2-(-3/2)}}} units long, and two other sides which are the two given lengths:

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{{{4*sqrt(2)=sqrt((x-(-3/2))^2+(y-(-5/2))^2)}}}
{{{4*sqrt(2)=sqrt((x+3/2)^2+(y+5/2)^2)}}}
{{{highlight(32=(x+3/2)^2+(y+5/2)^2)}}}_______We can make more use of this one.
which is a circle


{{{2*sqrt(5)=sqrt((x-9/2)^2+(y-(-5/2)))}}}
{{{highlight(20=(x-9/2)^2+(y+5/2)^2)}}}_______ .
also a circle


These two circles both contain the expression {{{(y+5/2)^2}}}, which give us a way to relate them.
{{{32-(x+3/2)^2=20-(x-9/2)^2}}}
{{{(x-9/2)^2-(x+3/2)^2-20 +32=0}}}
{{{x^2-(18/2)x+81/4-(x^2+(6/2)x+9/4)+12=0}}}
{{{x^2-9x+81/4-(x^2+6x+9/4)+12=0}}}
{{{-9x+81/4-6x-9/4+12=0}}}
{{{-15x+12+72/4=0}}}
{{{-15x+12+18=0}}}
{{{15x=30}}}
{{{highlight(x=2)}}}


What you now want to do, is go back to this same circle equation, solve it for y, and assign x=2 and find the value for y.  You will get two solutions for this y.  Whatever y is found to be, the two points wanted will be (2, y) and (2, -y).