the heck am I doing wrong here? (Orthogonal trajectory)

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PunkLuncheon

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CAL 2 here. Bear with me, first time posting.
The question is essentially: We choose arbitrary integers between 1 & 9 for a and b. Then find the family of equations which intersect orthogonally with this family of functions (the first line).
1603533646942.png1603534123551.png1603534165219.png


When I graph it it looks good at first glance. But for certain values (a,b,k) it's clearly off. I've gone over it multiple times.
If you see anything and can give me any hints I would appreciate it a lot!
 
1) I'm a little curious how you expect your family of orthogonal curves to be what you expect when they do not involve 'k'. Can you explain that?
2) Too many "C"s in that second-to-last equation, but you fixed it in the last equation.
3) I don't believe "clearly off". Can you prove it? That's a lot of work to throw out just because it doesn't SEEM right or doesn't LOOK as expected. Pick one of your points where it is "clearly off" and PROVE IT.
4) Very good presentation. Nicely legible.
5) Good work on the Absolute Value. Many will miss that.
6) Are your scales the same on both axes when it looks "clearly off"?
 
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tkhunny said:
1) I'm a little curious how you expect your family of orthogonal curves to be what you expect when they do not involve 'k'. Can you explain that?
2) Too many "C"s in that second-to-last equation, but you fixed it in the last equation.
3) I don't believe "clearly off". Can you prove it? That's a lot of work to throw out just because it doesn't SEEN right or doesn't LOOK as expected. Pick one of your points where it is "clearly off" and PROVE IT.
4) Very good presentation. Nicely legible.
5) Good work on the Absolute Value. Many will miss that.
6) Are your scales the same on both axes when it looks "clearly off"?
Click to expand...

Thanks for the reply!

& for your scaling tip: that certainly makes it look more "proper".

I will try to prove it perhaps more thoroughly. What I did so far was reduce each of the family and orth. traj. to explicit functions (y -> x), and derive both - and invert one. The derivative functions do not match (pic attached).
1603579278116.png
However, perhaps the derivative functions only match where the original and orth. functions intersect.

To answer your question of k, essentially our whole methodology is that if we replace k after we first derive, we are then able to integrate properly, and end up with a function which is orthogonal generally to the original function family (no matter the value of k). It usually works just fine!

Also I use mathcha - highly recommend.
 
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Indeed, Yes I realized that the inversed derivatives only intersect at the same x-value, and they did, so that confirms orthogonality. Thank you!
 
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