TITLE: Find the equation of the conic
QUESTION [0 upvotes]: Find the equation of the conic which passes through the points $(-1,-1)$ and also through the intersection of the conic $x^2+2xy+5y^2+x+16y+8=0$ with the straight lines $2x-y-3=0$ and $3x+y-3=0$. i failed to solve this question.the solution provided by the book is:$$34x^2+55xy+139y^2+13x+348y+233=0$$Can anyone help me to explain how to get the solution.
Thanks in advance.

REPLY [0 votes]: Each of straight lines cuts the given conic in 2 points, the fifth point $(-1,-1)$ is given. Plug the coordinates of these points into the equation $$ax^2+bxy+cy^2+dx+ey+f=0$$ of the resulting conic. Then solve the system of 5 linear equations with unknowns $a,b,c,d,e,f$ (one of the will be a parameter).
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