Explanation:
(A) : Given equation is,
\(3 x^{2}+10 x y+3 y^{2}+16 y+k=0\)
Comparing the given equation with
\(a x^{2}+2 h x y+b^{2}+2 g x+2 f y+c=0\).
We get, \(\mathrm{a}=3, \mathrm{~h}=5, \mathrm{~b}=3, \mathrm{~g}=0, \mathrm{f}=8, \mathrm{c}=\mathrm{k}\).
Now, given equation represents a pair of lines.
\(\therefore \quad \mathrm{abc}+2 \mathrm{fgh}-\mathrm{af}^{2}-\mathrm{bg}^{2}-\mathrm{ch}^{2}=0\)
\(\therefore(3)(3)(\mathrm{k})+2(8)(0)(5)-3(8)^{2}-3(0)^{2}-\mathrm{k}(5)^{2}=0\)
\(\therefore 9 \mathrm{k}+0-192-25 \mathrm{k}=0 \Rightarrow 16 \mathrm{k}=-192 \Rightarrow \mathrm{k}=-12\)