如图,已知抛物线y=ax2-2ax+b与x轴交于A、B(3,0)两点,与y轴交于C,且OC=3OA,

(1)抛物线y=ax^2-2ax+b的对称轴是x=1,与x轴交于B(3,0),A(-1,0),
则y=a(x+1)(x-3),
它与y轴交于C(0,-3a),由OC=3OA=3,
∴a=土1,
∴抛物线的解析式是y=x^2-2x-3,或y=-x^2+2x+3.
(2)对于抛物线y=x^2-2x-3,C(0,-3),其顶点D为(1,-4),设P(p,p^2-2p-3),p>1,
△PDC为等腰三角形,分3种情况：
1)PD=CD,p=2,P(2,-3);
2)PC=PD,p^2+(p^2-2p)^2=(p-1)^2+(p^2-2p+1)^2,
2p-1=2p^2-4p+1,2p^2-6p+2=0,p^2-3p+1=0,p>1,
∴p=(3+√5)/2,P((3+√5)/2,(-5+√5)/2);
3)CP=CD,p^2+(p^2-2p)^2=2,
p^4-4p^3+5p^2-2=0,
(p-1)(p^3-3p^2+2p+2)=0,无大于1的根.
对于抛物线y=-x^2+2x+3,留给您练习.可以吗?