Inequality $((4000 - x)^2 - (3200 - x)^2) > ((2100 - y)^2 - (4000 - y)^2)$

Question

I've been working with a distance formula and trying to come up with an efficient way of figuring out the range of $y$. Here's the formula:

$((4000 - x)^2 - (3200 - x)^2) > ((2100 - y)^2 - (4000 - y)^2)$

Basically, I will always know the $x$ value beforehand. I want an efficient way that when given $x$, will produce the range of $y$ that will make this inequality true.

This is for a program I'm coding, where simply going through all possible $y$ values is way too slow.

When I go on a site like Desmos, it plots it so quickly! I'd appreciate any help. Thanks!

I will add link to a related question: Equation of a line that passes halfway between two points (in other words, divides the space) (You're basically asking whether the point belongs to one of the half-planes given by the bisector.) — Martin Sleziak, Jul 08 '21 at 06:13

score 4 · Accepted Answer · answered Jul 07 '21 at 07:48

4

So you have $C > ((2100 - y)^2 - (4000 - y)^2)$
Thus $C > (2100^2 - 4000^2 + 3800y)$
$C + 4000^2 - 2100^2 > 3800y$
$y < \frac{C + 11590000}{3800}$ are values of y which satisify the inequality

answered Jul 07 '21 at 07:48

egglog

1,674

This is great, the steps were a good help too. Thanks for the solution! I don't know where my head was. – zapshe Jul 07 '21 at 07:57

score 4 · Answer 2 · answered Jul 07 '21 at 07:48

4

Expanding both sides, we obtain

$$1600(3600 - x) > 3800(y - 3050),$$

which simplifies to

$$y < -\frac{2}{19}(4x - 43375). $$

answered Jul 07 '21 at 07:48

Ekesh Kumar

3,500

Thanks a lot, seeing this now makes me wonder where my head was. Greatly appreciated. – zapshe Jul 07 '21 at 07:54

Inequality $((4000 - x)^2 - (3200 - x)^2) > ((2100 - y)^2 - (4000 - y)^2)$

2 Answers2