The formula involved is (two fixed point, ratio):
\(\dfrac{(x-x_1)^2+(y-y_1)^2}{(x-x_2)^2+(y-y_2)^2}=\dfrac{m^2}{n^2}\)
Substitute the value of the coordinates \(A(2,0)\) and \(B(4,-1)\) with ratio \(AP:PB=1:2\) into the formula:
\(\dfrac{AP}{PB}=\dfrac{1}{2}\)
\(\dfrac{(x-2)^2+(y-0)^2}{(x-0)^2+(y+2)^2}=\dfrac{1}{2^2}\)
Expand and simplify the resulting equation:
\(4[(x-2)^2+y^2]=x^2+(y+2)^2 \\ 4(x^2-4x+4)+4y^2=x^2+(y^2+4y+4) \\ 4x^2-16x+16+4y^2=x^2+y^2+4y+4 \\ 3x^2+3y^2-16x-4y+12=0\)
The equation of the locus of a moving point \(P\) is:
\(3x^2+3y^2-16x-4y+12=0\)
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