Let O 0,0 and A 0,1 be two fixed points. Then, the locus of a point P such that the perimeter of Δ A O…

$\Rightarrow PO+PA=3\mathrm{ }$

$\Rightarrow$Locus of $P$ is ellipse with foci at $O$ & $A$ and major

axis $\left(2b\right)=3$

Distance between foci $=2be=1\Rightarrow e=\frac{1}{3}$

$\Rightarrow$Minor axis $2a=2b\sqrt{1-e^2}=3.\frac{2\sqrt{2}}{3}=2\sqrt{2}$

$\Rightarrow$ Equation of Locus of $P$ is :

$\frac{x^2}{8}+\frac{{\left(y-\frac{1}{2}\right)}^2}{9}=\frac{1}{4}$

By simplifying the above equation, we get

$9x^2+8y^2-8y=16$