Answer : $5x^2+5y^2+180=0$
Given that the distance of the set of points from $(0,4)$ is $\large\frac{2}{3}$ of the distance from the line $y=9$
Hence $\sqrt{(x-0)^2+(y-4)^2}=\large\frac{2}{3}$$\sqrt{(x-0)^2+(y-9)^2}$
Squaring on both sides we get,
$x^2+(y-4)^2=\large\frac{4}{9}$$\big[(x^2)+(y-9)^2\big]$
$9(x^2+y^2-8y+16)=4x^2+4(y^2-18y+81)$
On simplifying we get,
$5x^2+5y^2+180=0$
This is the required equation of the set of points.