If $d_1,d_2(d_1<d_2)$ denote the lengths of the perpendicular from the point (2,3) on the lines given by $15x^2+31xy+14y^2=0$, then $d_1^2-d_2^2+\frac{1}{74}+\frac{1}{13}=$

The lines represented by $15x^2+31xy+14y^2=0$ are $5x+7y=0$ and $3x+2y=0$

length of perpendicular from (2,3) on these lines are $d_1=\frac{31}{\sqrt{74}},d_2=\frac{12}{\sqrt{13}}$

$\therefore d_1^2+\frac{1}{74}-d_2^2+\frac{1}{13}$

$=\left(\frac{{\displaystyle 961}}{{\displaystyle 74}}+\frac{{\displaystyle 1}}{{\displaystyle 74}}\right)-\left(\frac{{\displaystyle 144}}{{\displaystyle 13}}-\frac{{\displaystyle 1}}{{\displaystyle 13}}\right)=13-11=2$