This topic has come up on PHD2 forum. The answer to your question involves more than just the focal lengths now that cameras and software are the norm for guiding.
The OLD rule of thumb when manual guiding was the only game in town was the guide scope focal length had to be at least 1/2 that of the imaging scope. With the advent of autoguiding, the image scale of the two cameras comes into play.
Bruce Waddington posted some comments about requirements for guide scope and cameras on the PHD Stark Labs forum and copied to PHD2. Note the Stark Labs forum is no longer available
"Hi, this is Bruce. I posted my response on the Stark Labs forum last night. As I said there, what probably matters more is the relationship between the two image scales rather than two focal lengths – I think you already knew that. The goal is for the tracking/guiding to be seeing-limited, you want to keep any centroid error well below that. Beyond the simple stuff I described, it becomes a matter of your seeing conditions, the sensitivity of the guide camera and how faint the guide star is, the size of the star disks on your main system, etc. I don’t think there’s any cookbook answer for this. And if you’re imaging at long focal lengths (fine image scales), the issues of differential flexure will usually take over and most of this other stuff becomes irrelevant."
The following was part of the same thread
Bruce’s approach can be translated into a similar rule of thumb as the historical value with respect to focal length of the guide scope, if we accept a couple assumptions. This reduces the number of calcs needed.
GP = Guide camera pixel size in microns
GF = Guide camera focal length in mm
IP = Image camera pixel size
IF = Image camera focal length
1. Centroid accuracy of 0.2 pixels for the guiding software
2. Movement of less than 1 pixel on the main image is the acceptable upper limit
Guider image scale = GP * 206.265/GF
Imager image scale = IP * 206.265/IF
Acceptable Movement on main image = 1 pixel = GP * 206.265/GF * 0.2
IP * 206.265/IF
Rule of thumb
Guider focal length > 0.2*(GP/IP)*IF
Bruce’s first example
Guider focal length > 0.2*(5.3/7.3)*1680 = 229
The 350 f.l. guidescope is OK
Bruce’s second example
Guider focal length > 0.2*(5.3/7.3)*2540 = 354
The 350 f.l. guiderscope is marginal