SOLUTION: Find the equation that is satisfied by the coordinates of each point. A rod PQ of the length 12 moves so that P is always on the Y-axis and Q always on the X-axis. A point M on PQ

Algebra ->  Formulas -> SOLUTION: Find the equation that is satisfied by the coordinates of each point. A rod PQ of the length 12 moves so that P is always on the Y-axis and Q always on the X-axis. A point M on PQ       Log On


   

Question 1131594: Find the equation that is satisfied by the coordinates of each point. A rod PQ of the length 12 moves so that P is always on the Y-axis and Q always on the X-axis. A point M on PQ is 2/3 of the way from P to Q. What equation is satisfied by the coordinates of M?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
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From the length of the rod, you know,
%28x-0%29%5E2%2B%280-y%29%5E2=12%5E2
x%5E2%2By%5E2=144
and then you know that for the x coordinate,
%28a-0%29%2F%28x-0%29=2%2F3
a%2Fx=2%2F3
3a=2x
x=%283%2F2%29a
and for the y coordinate,
%28b-y%29%2F%280-y%29=2%2F3
3%28b-y%29=-2y
3b-3y=-2y
y=3b
So plugging back into the length equation,
%28%283%2F2%29a%29%5E2%2B%283b%29%5E2=144
%289%2F4%29a%5E2%2B9b%5E2=144
highlight%289a%5E2%2B36b%5E2=576%29
Just to verify,
when x=0,a=0, the point Q is at the origin, PQ lies entirely on the y axis,
9a%5E2%2B36b%5E2=576
36b%5E2=576
b%5E2=4
Two solutions : b=2, b=-2
Two thirds from P to Q would be %282%2F3%29%2812%29=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)