Geometry or calculus? II

Calculus Level 5

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

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