C

    Jackson was driving back home along a road one evening. He just lost his job. It was winter and it was very cold, just like his feelings.

    An old lady with her car was standing on one side of the road. The lady looked very worried. She needed help. No one stopped to help her. Jackson stopped in front of her and got out of his car. Even when he put a smile on his face, the old lady still looked worried. Was he going to hurt her? He didn't look safe, and he looked poor and hungry.

    Jackson knew how she felt. He said." I'm here to help you, madam. Why don't you wait in my car? By the way, my name is Jackson."

    The lady thought for a while and then went into his car. There was something wrong with her car. Jackson went under her car to repair it. After a long time, he finished it. He got dirty and his hands hurt. Then the lady told him that her name was Annie. She asked Jackson how much she should pay for his help.

    Jackson never thought about getting money. This was not a job to him. This was helping someone in need. He told her that if she really wanted to pay him back, the next time she saw someone who needed help, she should give that person help.

识图并回答问题：

已知点P是双曲线$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a>0,b>0\right)$和圆$x^2+y^2=a^2+b^2$的一个交点，$F_1{F}_2$是双曲线的两个焦点，$\angle P{F}_2{F}_1=2\angle P{F}_1{F}_2$，则双曲线的离心率为(      )