If (x, y, z) be an arbitrary point lying on a plane P which passes through the point (42,0,0) , (0, 42,0) ,(0,0,42), then the value of expression
$3+\frac{\mathrm{x}-11}{(\mathrm{y}-19{)}^2(\mathrm{z}-12{)}^2}+\frac{\mathrm{y}-19}{(\mathrm{x}-11{)}^2(\mathrm{z}-12{)}^2}+\frac{\mathrm{z}-12}{(\mathrm{x}-11{)}^2(\mathrm{y}-19{)}^2}-\frac{\mathrm{x}+\mathrm{y}+\mathrm{z}}{14(\mathrm{x}-11)(\mathrm{y}-19)(\mathrm{z}-12)}$

Plane passing through $(42,0,0),(0,42,0),(0,0,42)$
From intercept from, equation of plane $\mathrm{x}+\mathrm{y}+\mathrm{z}=42$
$\Rightarrow (\mathrm{x}-11)+(\mathrm{y}-19)+(\mathrm{z}-12)=0$
Let $\mathrm{a}=\mathrm{x}-11,\mathrm{b}=\mathrm{y}-19,\mathrm{c}=\mathrm{z}-12$
$\mathrm{a}+\mathrm{b}+\mathrm{c}=0$
Now, given expression is
$\begin{array}{l}3+\frac{\mathrm{a}}{{\mathrm{b}}^2{\mathrm{c}}^2}+\frac{\mathrm{b}}{{\mathrm{a}}^2{\mathrm{c}}^2}+\frac{\mathrm{c}}{{\mathrm{a}}^2{\mathrm{b}}^2}-\frac{42}{14\mathrm{abc}}\\ 3+\frac{{\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3-3\mathrm{abc}}{{\mathrm{a}}^2{\mathrm{b}}^2{\mathrm{c}}^2}\end{array}$
If $\mathrm{a}+\mathrm{b}+\mathrm{c}=0\Rightarrow {\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3\mathrm{abc}\Rightarrow 3$