Weakness in knees

A beginning student has an existing weakness in both knees, which is sometimes aggravated by her yoga practice, especially lunges, and particularly within Surya Namaskar. I am quite vigilant about ensuring correct alignment of her knees while she is in class but I am concerned that her competitive (and not highly conscious) nature may cause her to damage her knees while practicing yoga. What is the safest way to strengthen her knees without putting them at further risk?

There are many factors for knees. Your question is vague and I cannot answer it. Often difficulty with knees comes from hip and/or sacroiliac misalignments or lack of hip range of motion, which are often symptom free. To answer this question, I need an assessment of what is weak and what motions are limited. How to do this is in my book. I did a 2-part article on knee anatomy and therapy. You can get copies of my articles on yoga therapy for knees and shoulders reprints are available from me for $5 or from Yoga International Magazine.

Sounds like your greatest concern is not your teaching but her competitive mind. For that one can encourage ujjaye pranayama with a longer exhalation. After she is doing that successfully then you can go to next step or tease her that there is a more advanced technique but she will have to practice until her exhalation is 50% longer than inhalation. The next recommendation is to have her do short periods of running by breathing only through her nostrils. Over time this will increase strength of her lungs and make her more efficient in running. Details about this training for athletes are in Body, Mind and Sport by Dr. John Douillard, DC.