ASHINGTON — Authorities are investigating whether Mitt Romney’s private email account was hacked. On Tuesday, the website Gawker reported that an anonymous hacker had signed into Romney’s personal Hotmail account. Gawker reported that the hacker guessed the answer to a security question about Romney’s favorite pet in order to gain access to the account and change the password. The anonymous hacker told the website that Romney’s account on DropBox, a file sharing service, also was compromised.