The orthocentre of the triangle and a parabola

[LCKurtz]

Well, my interpretation and solution do not imply that the orthocenter MUST be inside triangle, although it certainly CAN be inside. My solution in fact has the orthocenter ON the triangle (and on the origin). The interpretation here is that they are saying there ARE three point that have one of the possible solutions as the correct answer and that's what my solution does.

We aren't disagreeing about the mathematics, only the interpretation of the problem. With your interpretation, which I don't think is reasonable, all the answers are correct.

If the points are $(0,0),(a^2,a),(a^2,-a)$ then:
If $a=1$ the orthocenter is at the origin and vertex. I guess that is what your picture is supposed to represent.
If $a = \frac {\sqrt 5} {2}$ the orthocenter is at $(\frac 1 4, 0)$, which is the focus.
If $a=\sqrt 2$ the orthocenter is at $(1,0)$.

No reason to prefer one answer over the others then.

 

[Raghav Gupta]

If $a = \frac 5 {\sqrt 2}$ the orthocenter is at $(\frac 1 4, 0)$, which is the focus.

Should be $a = \frac{\sqrt{5}}{2}$
Anyways I knew the answer that there is no answer because answer to be chosen should be one from options in much earlier posts but that phinds guy was confusing me.
Thanks for showing the calculations.
By the way thanks to all for a discussion.

 

[phinds]

Should be $a = \frac{\sqrt{5}}{2}$
Anyways I knew the answer that there is no answer because answer to be chosen should be one from options in much earlier posts but that phinds guy was confusing me.
Thanks for showing the calculations.
By the way thanks to all for a discussion.

I wish you would stop calling me "that phinds guy". It is quite rude.

 

[Raghav Gupta]

Sorry, Mr phinds or Sir phinds. I earlier thought guy was a nice word.

In a virtual world we don't know the identity of a person actually and that avatar of you gives me some other feeling.

The problem arises for me when more then two persons are involved and we have to refer to someone in third person.

Recently looked at your profile information and you are a very experienced person.
Thanks Mr.Phinds for the discussion.

 

[LCKurtz]

Should be $a = \frac{\sqrt{5}}{2}$

Yes. Thanks for catching that typo. I will correct it.

 

[Ray Vickson]

Sorry, Mr phinds or Sir phinds. I earlier thought guy was a nice word.

In a virtual world we don't know the identity of a person actually and that avatar of you gives me some other feeling.

The problem arises for me when more then two persons are involved and we have to refer to someone in third person.

You can just say "phinds in post #xx" or something similar. In principle, you may not know if a user is actually a "guy" at all!

Recently looked at your profile information and you are a very experienced person.
Thanks Mr.Phinds for the discussion.