Can you please help me replacate this method for finding the common tangent

No.13487977 ViewReplyOriginalReport
Im beyond tilted. Please math anons I seek your help.

https://math.stackexchange.com/questions/2526312/find-tangent-points-for-common-tangent-to-two-curves-described-with-parametric-e

How can I replicate the first method for
a=0.5
b=1
p=1
P=(1,1.75) (same)
P1=(2,2) (same)

Please teach me