A pipe can fill a cistern in 9 hours.

Time taken by the pipe to fill the cistern $=9$ hours

Part of the cistern filled in one hour $=\frac{1}{9}$

Suppose the leak empties the full cistern in x hours.

Part of the cistern emptied in one hour $=-\frac{1}{x}$ (negative sign implies a leak)

Time taken by the cistern to fill completely due to the leak $=10$ hours

Part of the cistern filled in one hour due to the leak $=\frac{1}{10}$

$\therefore \frac{1}{10}=\frac{1}{9}-\frac{1}{x}$

$\Rightarrow \frac{1}{x}=\frac{1}{9}-\frac{1}{10}=\frac{1}{90}$

$x=90$ hours

Therefore, the leak will empty a full cistern in 90 hours.