Would a transabdominal-pelvic ultrasound detect an ectopic pregnancy at 9 weeks?

Maybe: The accuracy of a transabdominal pelvic ultrasound will depend on several factors: the level of the blood betaHCG tests, (not the LMP or estimated weeks of pregnancy) the skills and experience of the sonographer, (Certified Registered Technician) the quality and resolution power of the ultrasound machine, its number of MegaHertz, the location of the ectopic pregnancy (tubal,ovarian, cervical..)

Possibly, but: a transvaginal ultrasound would have a better chance. Note that the role of US is primarily to document the absence of an intrauterine pregnancy in the presence of rising/elevated pregnancy hormones to help diagnose ectopic pregnancy. The ectopic itself isn't always visualized, and doesn't have to be, to make the diagnosis