Question 1131594
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 *[illustration D9.png].
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From the length of the rod, you know,
{{{(x-0)^2+(0-y)^2=12^2}}}
{{{x^2+y^2=144}}}
and then you know that for the x coordinate,
{{{(a-0)/(x-0)=2/3}}}
{{{a/x=2/3}}}
{{{3a=2x}}}
{{{x=(3/2)a}}}
and for the y coordinate,
{{{(b-y)/(0-y)=2/3}}}
{{{3(b-y)=-2y}}}
{{{3b-3y=-2y}}}
{{{y=3b}}}
So plugging back into the length equation,
{{{((3/2)a)^2+(3b)^2=144}}}
{{{(9/4)a^2+9b^2=144}}}
{{{highlight(9a^2+36b^2=576)}}}
Just to verify, 
when {{{x=0}}},{{{a=0}}}, the point Q is at the origin, PQ lies entirely on the y axis,
{{{9a^2+36b^2=576}}}
{{{36b^2=576}}}
{{{b^2=4}}}
Two solutions : {{{b=2}}}, {{{b=-2}}}
Two thirds from P to Q would be {{{(2/3)(12)=8}}} since P is at (0,12) then M should be at (0,12-8)=(0,4), which is consistent to the answer from the equation. P could also be at (0,-12). Similary 2/3 from P to Q would be (0,-12+8)=(0,-4).
You can do the same check when PQ is entirely on the x-axis just to verify ({{{b=0}}})