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Let (x,y) be any arbitrary point on the ellipse.
distance of (x,y) from (0,4), d₁=  \sqrt{(x-0)^2+(y-4)^2} =\sqrt{x^2+(y-4)^2

distance of (x,y) from the line y-9=0, d₂ =  \frac{(y-9)}{1}=y-9

(Note that distance of a point (X,Y) from a line ax+by+c=0 is \frac{aX+bY+c}{ \sqrt{a^2+b^2} }

given that d₁ = 2/3 d₂

\sqrt{x^2+(y-4)^2}= \frac{2}{3} (y-9) \\ \\ x^2+(y-4)^2= \frac{4}{9} (y-9)^{2} \\ \\ 9(x^2+(y-4)^2)=4( y-9)^{2} \\ \\ 9 (x^{2} +y^{2} -8y+16)=4( y^{2} -18y+81)

9x^{2} +9y^{2} -72y+144=4y^{2} -72y+324 \\  \\9x^{2} +5y^{2}-180=0

This is the required equation of the ellipse.

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