If the locus of the point, whose distances from the points \((2,1)\) and \((1,3)\) are in the ratio \(5: 4\), is \(a x^2…

Let point be \(\mathrm{P}(\mathrm{x}, \mathrm{y})\)

As per the question:

\(\frac{\sqrt{{\left(x-2\right)}^{2}+{\left(y-1\right)}^{2}}}{\sqrt{{\left(x-1\right)}^{2}+{\left(y-3\right)}^{2}}}=\frac{5}{4}\)

\(\Rightarrow \frac{{\left(x-2\right)}^{2}+{\left(y-1\right)}^{2}}{{\left(x-1\right)}^{2}+{\left(y-3\right)}^{2}}=\frac{25}{16}\)

\(\Rightarrow 16\left({x}^{2}-4x+4+{y}^{2}-2y+1\right)\)\(=25\left({x}^{2}-2x+1+{y}^{2}-6y+9\right)\)
\(\Rightarrow 9 x^2+9 y^2+14 x-118 y+170=0\)

Comparing with \(a x^2+b y^2+c x y+d x+e y+170=0\)

\(a=9, b=9, c=0, d=14, e=-118\)

Now,
\(a^2+2 b+3 c+4 d+e\)\(=81+18+0+56-118\)
\(=155-118=37\)