Advise on Contacting Potentialy Unsuspecting Birth Siblings

My mother is an adopted child and seeking to contact her birth relatives. In short, my mother was born out of wedlock and her birth mother married several years later to someone other than her birth father. Her birth mother went on to raise a family of four children and has since passed. We have the number (or at least believe we do) of one of her half siblings and my mother has requested that I make the initial contact. In going forward, we acknowledge that the fact that there exists the very real possibility that my mothers birth mother never told any of her children about my mom. We both want to be sensitive to the potential impact this could have on the family, especially if they were never told about my mother. If anyone has any insight, guidance, suggestions (all of the above) as to how to best approach this initial contact and/or other things to be mindful of, it would be greatly, greatly appreciated. Below is more detail on my mother's story if anyone cares to read further: Up until her (adoptive) mothers (my grandmother) death in 1999, she never spoke of her adoption much, but mostly out of lack of interest and, as I found out later, respect for her (adoptive) parents. After the passing of her last parent, she came across her adoption papers. However, any pertinent names or information about her birth parents were blacked out. Over the next number of years she had the records released and finally had the name of her birth mother and father including additional details surrounding her adoption. In short, she was born out of wedlock, the father was likely not aware that he had fathered a child and the mother was giving her up for adoption. My Mom did a little research about her history but did not show much interest in tracking down her birth family. I, being intrigued with genealogy in general but most certainly that of my own family, asked my mother if she mind if I do some additional research about her history to which she obliged. Over time I have done some internet research and discovered that a woman with the same name in approximate age to what her birth mother would be at the time, had passed fairly recently (2008). Life was busy at the time and I never pressed further to find out more info or confirm if this was her birth mother, until recently. I purchased a record of obituary for this person and found that she had later remarried and raised a family of 4 children. However, citations of the deceasedҒs relatives were a match to that of the names of relatives of (and same relation to) the birth mother listed in the adoption papers. Listed in the obit were also the names of all of her children. Through name searches in various places on the internet, I have come up with a live phone number to who would be my mothers next oldest sibling (a half-sister). Of course I passed along this information to my mother. She seemed a bit intrigued with my findings but didnt indicate that she was interested in calling. A number of weeks later, she was still unsure if she wanted to make contact with her birth relatives but asked if I would be willing to call on her behalf if she decided to. My answer was ғof course. A little while later she indicated to me that she wanted to make contact citing (as she had previously as a reason for doing so) that she wanted to know more about her medical history for not only her sake, but also for mine and my childrenԒs benefit. Although she doesnt say so, I get the sense she is very intrigued (as I am) and wants to know about her extended family. So this is the crossroads at which we stand. I have called the number I found once only to verify it as a live number. I got an answering machine, but no names given to corroborate that it was the number I was suspecting.