How to combine a quadratic curve and a line to form an equation with only quadratic terms?

[mmamathes (Bill Blair)]

This is my own try at making this work with the substitution method. It’s manageable if you know the equation well, but the steps get really tedious with a complex quadratic equation.

L = m x + n y;
ellipseOriginal = (x + x0)^2/a^2 + (y + y0)^2/b^2 == 1; 
ellipseHomogenized1 = 
 ellipseOriginal /. {x0 -> x0*L, y0 -> y0*L, 1 -> L^2}

How to use code to automatically generate such a homogeneous quadratic equation by combining the linear equation and the quadratic curve?

[mmamathes (Bill Blair)]

[wolframcommunity (EDITORIAL BOARD)]

Welcome to Wolfram Community!
Please provide your efforts in the form of the Wolfram Language code. This will make it easier for other members to help you. Check several methods available to include your code in the rules http://wolfr.am/READ-1ST

[mmamathes (Bill Blair)]

I have also added my own attempted solutions.

[anon21378091]

Homogenize: w^2 \, f(x/w, y/w); get in the form you want; then replace w by mx+ny.