Doctors feel sure to have a try on the belief ____ there's life,there's hope.

填that where
that是跟在belief后面引导同位语从句的，后面where there's life,there's hope是同位语从句的内容

希望满意

补充：that是连接词啊，就是引导从句的成分
但是where是和后面的句子组成从句的，是从句内的成分之一，不是连接词

再补充：只有一个连接词，就是that
where是从句里的副词
有句谚语，叫Where there is a will, there is a way.(有志者，事竟成)，是同样的用法，在本身独立的句子里，就用到了副词where。
然后句子作为整体，前面加个that，放到主句后当同位语从句。
where在这个位置上出现纯粹是偶然

that where,,

但是这里的that where是分开的,,不是一个短语

that表同位语从句,,

where是地点状语引导词,,同where there is a will,there is a way.
句式

答案是 that where
Doctors feel sure to have a try on the belief ____ there's life,there's hope.
“医生们确切地想要实践这个信念，“有生命，就有希望””the belief 是名词，后接that引导的同位语从句，从句中由where引导there's life做there's hope的状语。
类似的用法还有“有志者，事竟成”“Where there is a will, there is a way”

that 同位语从句