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 $\dfrac{d+e+f-2}{a+b+c}$ .