c
Let $\Delta = \begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}}\\
{{{\left( {a + \lambda } \right)}^2}}&{{{\left( {b + \lambda } \right)}^2}}&{{{\left( {c + \lambda } \right)}^2}}\\
{{{\left( {a - \lambda } \right)}^2}}&{{{\left( {b - \lambda } \right)}^2}}&{{{\left( {c - \lambda } \right)}^2}}
\end{array}$
apply ${R_2} \to {R_2} - {R_3}$
$\Delta = \begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}}\\
{{{\left( {a + \lambda } \right)}^2} - {{\left( {a - \lambda } \right)}^2}}&{{{\left( {b + \lambda } \right)}^2} - {{\left( {b - \lambda } \right)}^2}}&{{{\left( {c + \lambda } \right)}^2} - {{\left( {c - \lambda } \right)}^2}}\\
{{{\left( {a - \lambda } \right)}^2}}&{{{\left( {b - \lambda } \right)}^2}}&{{{\left( {c - \lambda } \right)}^2}}
\end{array}$
$ = \begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}}\\
{4a\lambda }&{4b\lambda }&{4c\lambda }\\
{{{\left( {a - \lambda } \right)}^2}}&{{{\left( {b - \lambda } \right)}^2}}&{{{\left( {c - \lambda } \right)}^2}}
\end{array}$
($\because $ ${\left( {x + y} \right)^2} - {\left( {x - y} \right)^2} = 4xy$)
Taking out $4$ common form $\,{R_2}$
$ = 4\begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}}\\
{a\lambda }&{b\lambda }&{c\lambda }\\
{{a^2} + {\lambda ^2} - 2a\lambda }&{{b^2} + {\lambda ^2} - 2b\lambda }&{{c^2} + {\lambda ^2} - 2c\lambda }
\end{array}$
Apply ${R_3} \to \left[ {{R_3} - \left( {{R_1} - 2{R_2}} \right)} \right]$
$ = 4\begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}}\\
{a\lambda }&{b\lambda }&{c\lambda }\\
{{\lambda ^2}}&{{\lambda ^2}}&{{\lambda ^2}}
\end{array}$
Taking out $\lambda $ common from ${{R_2}}$ and from ${{R_3}}$.
$ = 4\lambda \left( {{\lambda ^2}} \right)\begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}}\\
a&b&c\\
1&1&1
\end{array}$
$ = k\lambda \begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}}\\
a&b&c\\
1&1&1
\end{array}$
$ \Rightarrow k = 4{\lambda ^2}$