$解:如答图②，当0\lt a≤3时$
$∵点P与点P_{1}关于y轴对称,P(-a,0),∴P_{1}(a,0)$
$设P_{2}(x_{1},0),又∵点P_{1}与点P_{2}关于直线l对称$
$∴\frac{x_{1}+a}{2}=3，即x_{1}=6-a$
$∴P_{2}(6-a,0),则P_{1}P_{2}=6-a-a=6-2a$
$如答图③当a＞3时$
$∵点P与点P_{1}关于y轴对称,P(-a,0),∴P_{1}(a,0)$
$设P_{2}(x_{2},0),又∵点P_{1}与点P_{2}关于直线l对称$
$∴\frac {x_{2}+a}{2}=3,即x_{2}=6-a$
$∴P_{2}(6-a,0)$
$则P_{1}P_{2}=a-(6-a)=2a-6$
$综上所述，当0\lt a≤3时,P_{1}P_{2}=6-2a$
$当a＞3时，P_{1}P_{2}=2a-6$
