Vertices of a variable triangle are
$$(3,4)\\ (5\cos\theta,5\sin\theta) \\ (5\sin\theta,-5\cos\theta) $$
where $\theta \in \mathbb R$. Given that the orthocenter of this triangle traces a conic, evaluate its eccentricity.
I was able to find the locus after three long pages of cumbersome calculation. I found the equations of two altitudes of this variable triangle using point slope form of equation of a straight and then solved the two lines to get the orthocenter. However, the equation turned out to be of a non standard conic. I evaluated its $\Delta$ to find that it's an ellipse, but I don't know how to find the eccentricity of a general ellipse.
Moreover, there must be a more elegant way of doing this since the questions in my worksheet are to be solved within $5$ to $6$ minutes each but this took way long using my approach.