0
$\begingroup$

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

Thank you very much

$\endgroup$
2
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$
0
0
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.