Geometry or calculus? II

Calculus Level 5

Given a ellipse with equation 2x2+3y2+xy5=02x^2+3y^2+x-y-5=0 and a point P(3,1)P(3,-1), there are two lines passing through PP that are also tangent to the ellipse. One line is represented by the equation ax+byc=0ax+by-c=0 and the other one is dxeyf=0dx-ey-f=0, where a,b,c,d,e,f>0a,b,c,d,e,f>0, gcd(a,b,c)=1\gcd(a,b,c)=1 and gcd(d,e,f)=1\gcd(d,e,f)=1. Find d+e+f2a+b+c\dfrac{d+e+f-2}{a+b+c}.

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