Find the Locus of the Orthocenter

Vertices of a variable triangle are $(3,4)\phantom{\rule{0ex}{0ex}}(5\mathrm{cos}\theta ,5\mathrm{sin}\theta )\phantom{\rule{0ex}{0ex}}(5\mathrm{sin}\theta ,-5\mathrm{cos}\theta )$ 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 Δ 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.

Vertices of a variable triangle are $(3,4)\phantom{\rule{0ex}{0ex}}(5\mathrm{cos}\theta ,5\mathrm{sin}\theta )\phantom{\rule{0ex}{0ex}}(5\mathrm{sin}\theta ,-5\mathrm{cos}\theta )$ 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 Δ 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.