Hi pulkit
I think the Mage_Captcha was disabled, so please check the Mage_Capcha.xml file to ensure it was enabled.
If it is enabled but not show configurations in backend, you need download source code of magento 1.9 then compare if the extension is missig any files in the Mage_Captcha extension.
By the way, you can reference this extension: Studioforty9_Recaptcha, I tried to intalled this extension on my site, it is working well and easy to customize.
Hope that help and good luck!