STEP geometry problem. Is division by zero sometimes irrelevant / ok to be ignored?

https://pmt.physicsandmathstutor.com/download/Maths/STEP/Papers/2006%20STEP%202.pdf

Question 7...

The final part asks whether the result "holds". By "the result", I assumed it meant the given tangent equation, and when substituting sin(alpha) = 0 into the term for the intersect, b(cosec(alpha)) I suspected that because the term, and hence part of the result was undefined, there would be a general failure to satisfy the requirement that the result "hold", in the strictest sense (because part of the given expression has become undefined). However, I'm guessing it's considered sufficiently obvious that the "result" required to "hold" refers primarily to the existence of a vertical tangent, and here the intersect term is simply inactive and can be ignored. My question is, more generally, when can an undefined term be ignored and simply counted as non-existent without invalidating the whole of the expression which it belongs to?

Thank you.

(edited 5 years ago)

Reply 1

DFranklin

Original post by jameshyland29

I don't think that's what they mean. The result would typically be the last thing proved - as that is effectively the culmination of the work so far.

So in this case, "the result" is: "the line PQ is tangent to the above ellipse at the point given by tan(α/2) = k."

I suspected that because the term, and hence part of the result was undefined, there would be a general failure to satisfy the requirement that the result "hold", in the strictest sense (because part of the given expression has become undefined). However, I'm guessing it's considered sufficiently obvious that the "result" required to "hold" refers primarily to the existence of a vertical tangent, and here the intersect term is simply inactive and can be ignored.

I don't recall details of the question, but in this particular case, it's pretty clear that:

(a) they've given you those restrictions on k to ensure that a "find the tangent in the form y = mx + c, and do some algebra" approach doesn't hit any division by zero questions.
(b) They want you to check whether the result still holds for two values of k where you do get division by zero questions.
(c) They are not expecting you to simply ignore division by zero or similar; I'd say this was fairly obvious just by the fact that they have specifically asked (b), but the phrase "determine by sketches..." makes it pretty explicit that you can't simply use the previous calculation.

My question is, more generally, when can an undefined term be ignored and simply counted as non-existent without invalidating the whole of the expression which it belongs to?

Pretty much never (*). There may be times when you can *justify* ignoring it (stuff is going towards infinity but getting divided by something going to infinity even further), but I can't think of a circumstance where you can simply ignore the issue.

[Exception: you sometimes get scenarios with infinite series where you might be told to ignore issues of convergence. In which case the sum of the series may technically be undefined but you can ignore that when doing things like comparing coefficients etc. ]

STEP geometry problem. Is division by zero sometimes irrelevant / ok to be ignored?

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