what is the area of intersection of $a<\frac{x}{y}<b$ and $c<\frac{y}{x}<d$?

Thank you very much

  • $\begingroup$ Have you drawn a picture? $\endgroup$ – Arthur Aug 7 '14 at 15:00
  • $\begingroup$ I tried but. There is a lot of specieal cases. $\endgroup$ – Boby Aug 7 '14 at 15:05

Since $y^2\gt 0$, multiplying everything by $y^2$ gives you $$a\lt \frac{x}{y}\lt b $$$$\iff ay^2\lt xy\lt by^2$$$$\iff ay^2-xy\lt 0\ \text{and}\ by^2-xy\gt 0$$$$\iff y(ay-x)\lt 0\ \text{and}\ y(by-x)\gt 0$$$$\iff "y\gt 0\ \text{and}\ ay-x\lt 0\ \text{and}\ by-x\gt 0"\ \text{or}\ "y\lt 0\ \text{and}\ ay-x\gt 0\ \text{and}\ by-x\lt 0"$$

Here, since we have $a\lt b$, there are several cases to consider :

(1) If $0\lt a\lt b$, then we have $$"y\gt 0\ \text{and}\ y\lt\frac xa\ \text{and}\ y\gt\frac xb"\ \text{or}\ "y\lt 0\ \text{and}\ y\gt\frac xa\ \text{and}\ y\lt\frac xb"$$

(2) If $0=a\lt b$, then we have $$"y\gt 0\ \text{and}\ x\gt 0\ \text{and}\ y\gt\frac xb"\ \text{or}\ "y\lt 0\ \text{and}\ x\lt 0\ \text{and}\ y\lt\frac xb"$$

(3) If $a\lt 0\lt b$, then we have $$"y\gt 0\ \text{and}\ y\gt\frac xa\ \text{and}\ y\gt\frac xb"\ \text{or}\ "y\lt 0\ \text{and}\ y\lt\frac xa\ \text{and}\ y\lt\frac xb"$$

(4) If $a\lt 0=b$, then we have $$"y\gt 0\ \text{and}\ y\gt\frac xa\ \text{and}\ x\lt 0"\ \text{or}\ "y\lt 0\ \text{and}\ y\lt\frac xa\ \text{and}\ x\gt 0"$$

(5) If $a\lt b\lt 0$, then we have $$"y\gt 0\ \text{and}\ y\gt\frac xa\ \text{and}\ y\lt\frac xb"\ \text{or}\ "y\lt 0\ \text{and}\ y\lt\frac xa\ \text{and}\ y\gt\frac xb"$$

You'll get the similar cases as above for the latter.


When looking at this problem you should consider the following:

  1. What are the possible values for $|\frac{x}{y}|$ and $|\frac{y}{x}|$
  2. What values can $a,b$ and $c,d$ take?

With this in mind, first $|\frac{x}{y}|$ can be a small value (i.e. < 1), in which case $|\frac{y}{x}|$ should be a large value ( i.e. >1). Or they may be equal, or the opposite may occur!

Now how could we express that with the intervals $(a,b)$ and $(c,d)$?

I think of it as a Venn Diagram (or the intersection of neighborhoods of the real line).


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