I'm not sure exactly what you mean by "determination," but your question does raise some important issues about the quality of intention that we bring to our yoga. I hope my thoughts will address your question.
My immediate response is to try to loosely define what I think of as the ideal kind of determination to bring to our practice. Sometimes it helps to define something by identifying what it isn't. For me, determination in yoga is not any version of stubbornness or will power or fortitude, qualities that are often considered positive and that we sometimes associate with determination. I try to keep such qualities out of my yoga practice, because I don't want to consciously bring anything onto my yoga mat that might make me aggressive or willful.
This doesn't mean that I want to be lazy or less than fully involved and engaged in my practice, but simply that I want to approach my yoga less forcefully. Yoga does require commitment: commitment to showing up on a regular basis, to being willing to gently try even when you feel tired or checked out, to being open to whatever presents itself, to responding to what is actually happening and not what you think should be happening. As I'm sure you have heard many times and in many forms, yoga is about being true to the present moment. This means developing our conscious ability to observe and participate in the present moment, and learning how to use our asana practice as a forum for experiencing and enhancing this awareness.
Being present, however, also means that we cannot come onto our yoga mat with an agenda. Often when we "make a determination" or are "determined," we become committed to a plan or an idea that obstructs our ability to be open to the present moment. Therefore, instead of approaching your asana with "determination," I encourage you to make a slight semantic shift and think of approaching your practice with equal parts patience and persistence. This way you still have committed energy, but in a somewhat less rigid form.