# Equation of a general conic from 3 points and the major axis

I have read that given 3 points on a conic and the equation ($ax+by+c=0$) of its major axis, we can write the equation of the conic ($Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$). I've seen it done by rotating the known axis to align with a coordinate axis and using the "axis-aligned" representation of an ellipse/hyperbola ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$), but I was wondering if there was a way to do it without using that form?

It seems as though to get the equation of the major axis given a conic in general form you simply construct the matrix of conic coefficients and compute the eigenvector corresponding to the largest eigenvalue, but how would one do the reverse?

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