How can I find properties of conic section (center, vertices, foci, etc.)?

Is there a built-in function that can compute properties of conic section from its equation? For example, provided the equation $\ (x-3)^2/25+(y-4)^2/16=1$ is it possible to find its center, foci, vertices, eccentricity, etc using built-in functions? PlaneCurveData is close to what I'm looking for, for an ellipse I need to specify a and b to compute foci or eccentricity:

In[117]:= PlaneCurveData["Ellipse", "CartesianEquation"][5, 4][x, y]
Out[117]= x^2/25 + y^2/16 == 1
In[119]:= PlaneCurveData["Ellipse", "Foci"][5, 4]
Out[119]= {{-3, 0}, {3, 0}}
In[126]:= PlaneCurveData["Ellipse", "Eccentricity"][5, 4]
Out[126]= 3/5


But I still cannot change the center of the ellipse.

• Somebody once developed an entire package for this, so the effort needed is nontrivial. Commented Aug 16, 2017 at 6:42

So Entity["PlaneCurve", "Ellipse"] just have two variables

Entity["PlaneCurve", "Ellipse"]["Variables"]


So you cannot adjust the center of the ellipse to solve your question by the Entity solution.

ResourceFunction["ConicProperties"][x^2/25 + y^2/16 == 1, {x, y}]


[Not a "true" answer so much as leveraging the work of others.]

One can use WolframAlpha for this.

WolframAlpha["properties of x^2/25+y^2/16==1"]


• I don't know why this code don't work, maybe I have missed somethingEntity["PlaneCurve", {EntityProperty["PlaneCurve", "CartesianEquation"] -> Function[{\[FormalX], \[FormalY]}, 1/25 (-3 + \[FormalX])^2 + 1/16 (-4 + \[FormalY])^2 == 1]}][ EntityProperty["PlaneCurve", "Eccentricity"]]
– yode
Commented Jan 26, 2022 at 9:47