I am working on the following problem:
I need to find the points above the line $l:y=-x$ for which it holds that the distance to $l$ is $\sqrt{2}$ greater than the distance to the origin.
I worked out a way to calculate the distance from a point to the line $l$:
$$
\sqrt{\frac{(x+y)^2}{2}}.
$$
And then obviously the distance from a point to the origin is:
$$
\sqrt{x^2+y^2}.
$$
I then tried to plot the function:
$$
\sqrt{\frac{(x+y)^2}{2}} = \sqrt{x^2+y^2} + \sqrt{2}
$$
in Desmos and wolframalpha and both plots gave me nothing…
I even tried to plot just:
$$
\sqrt{\frac{(x+y)^2}{2}} = \sqrt{x^2+y^2}
$$
for which I know the line $y=x$ should be a solution, but this didn't plot anything either.
I'm not sure what is causing this. Why are Desmos and wolframalpha not able to Plot the curves ? Any ideas?
Many thanks,
Hugo
Best Answer
PROBLEM :
There is 1 MAJOR Issue ( & 1 MINOR Issue ) with your formulation , which is why neither DESMOS nor Wolfram will give the necessary graph.
(MAJOR) Distance to Origin ( $A$ ) is always more than ( or Equal to ) Distance to the line ( $B$ ) , in the given Case.
Hence , when you add $\sqrt{2}$ to $A$ , you are making it even larger than $B$ & there is no Point which will work out.
You should rather add $\sqrt{2}$ to $B$ to make the Correct formulation.
This Image highlights that Issue :
(MINOR) You are using $\sqrt{(x+y)^2}$ when trying to get $B$ magnitude , which is a little round-about.
Instead , you should use the More Direct $|x+y|$ & check that.
OUTPUT :
When I made those Changes , I got the necessary graph :
I used Wolfram Online Tool here. DESMOS too will give that Plot without issues now.