It's a good idea, but if your product is a website, cookies and authentication in general are going to be a real challenge. Obviously you won't be able to share cookies between the site and the api. But you have the additional challenge of having your api domain treated as 3rd party and are thus subject to all the security measures and sandboxing that come with that.
And what if your product is just an api for use by developers? Now the api is part of your brand.
And what if your product is just an api for use by developers? Now the api is part of your brand.
It's just a bad situation all around.