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In problems involving coordinate geometry, a lot of the time the solution provided only considers the simplest case and treats it as though the results obtained from it would work for every case. For example, one question asks,

Find the curve for which the area of the triangle formed by the x-axis, the tangent line and radius vector of the point of tangency is equal to $a^2$.

Now the solution in the book uses the simplest case(apologies for the abysmal sketches);

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Here and in the following pics, P is the point of tangency. This first case is simple enough, but these are possible, too;

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And many more.

Now to find the areas of these triangles, the base has to be found using the subtangent and the abscicca, and each case has it differently; their magnitudes are added or subtracted according to the case.

I could write all of them down and solve the equations, but there probably is a more intuitive way by which the general case can be solved. I say this because I've seen multiple questions like this, and all of them are solved in the book's solutions using just the one case. Which means it's kind of a waste of time to evaluate separate cases, because they're all apparently equivalent.

So is there any such intuition? How can I use it?

EDIT: When you work out the above three cases for the point being in each quadrant, you get the base to be, respectively, $$x-\frac{y}{m}\\ x-\frac{y}{m}\\ \frac{y}{m}-x$$ for the first and fourth quadrants, and the exact negatives of these for the second and third quadrants(where x and y represent the coordinates of the point, and m the slope.). This is what I meant- they are all of almost the same form, and if you take the absolute value, all twelve values are the same. Is there any way to think about these kinds of situations without having to work out each case?

harry
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After running into multiple questions of this sort, it seems to me that in cases where working it out in your head may be cumbersome and time-consuming, the best way forward is to do treat the situation in a purely computational manner, without thinking about it graphically. Though such an intuitive approach would be ideal, there isn't one, and the only way out is to churn out every single case as in the question - which leaves plenty of room for error. Therefore, working with numbers and equations might be a better choice.

harry
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    Nevertheless, it can be helpful to rely on a particular graphical interpretation for a quick-and-dirty visualisation of the problem, as long as the eventual algebraic solution method is rigorous (in other words, as long as the algebraic setup does not depend on a quirk that is specific to the chosen case). – ryang May 27 '21 at 09:08
  • @RyanG: do you mean the rough idea you keep in your head while writing out the algebraic solution? That's what I was thinking, too- the only thing is, if you're not doing calculation with graphical intuition, it could get messy. For instance, I churned out the equations in my question using intuition and algebra. Just now, I tried going in blind; using (x, y) for the general curve point, a slope-point equation of the tangent involving the term 'dy/dx', and the x-intercept using that equation. This lead to a differential equation. (Contd below) – harry May 27 '21 at 09:19
  • (contd.from above) So using a bare-bones, very basic setup to work it out algebraically might get messy (the differential equation), but working it out intuitively might take time and you might miss out cases(as in the question). This seems to be how it goes; I assume you referred to the "bare-bones" idea? – harry May 27 '21 at 09:22
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    Yep, I was agreeing with and expanding on your point. A graphical sketch can even inform the algebraic solution by pointing out a separate case that might’ve been overlooked in the latter; e.g., when the given info results in two possible triangles where angle A is 30 & 150 respectively. – ryang May 27 '21 at 10:11
  • @RyanG: shouldn't the algebraic treatment - if done properly and rigorously - always give you all the answers? – harry May 27 '21 at 10:31
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    Yep- my choice of phrase was “overlooked” -) – ryang May 27 '21 at 10:54