(a) Let point $P$ $({x_1},{y_1})$

So,$\sqrt {{{({x_1} + 2)}^2} + y_1^2} = \frac{2}{3}\left( {{x_1} + \frac{9}{2}} \right)$

==> ${({x_1} + 2)^2} + y_1^2 = \frac{4}{9}{\left( {{x_1} + \frac{9}{2}} \right)^2}$

==> $9[x_1^2 + y_1^2 + 4{x_1} + 4] = 4\left( {x_1^2 + \frac{{81}}{4} + 9{x_1}} \right)$

==> $5x_1^2 + 9y_1^2 = 45$

==>$\frac{{x_1^2}}{9} + \frac{{y_1^2}}{5} = 1$,

Locus of $({x_1},\,{y_1})$ is $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1$, which is equation of an ellipse.