Answer
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Hint: -Take \[x,y,z\] common from column 1, column 2, and column 3 respectively.
Consider L.H.S
\[ \Rightarrow \left| {\begin{array}{*{20}{c}}
{ax}&{by}&{cz} \\
{{x^2}}&{{y^2}}&{{z^2}} \\
1&1&1
\end{array}} \right|\]
Take \[x,y,z\] common from column 1, column 2, and column 3 respectively.
\[ \Rightarrow \left| {\begin{array}{*{20}{c}}
{ax}&{by}&{cz} \\
{{x^2}}&{{y^2}}&{{z^2}} \\
1&1&1
\end{array}} \right| = xyz\left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{\frac{1}{x}}&{\frac{1}{y}}&{\frac{1}{z}}
\end{array}} \right|\]
Now multiply $xyz$ in the third row of the determinant.
\[ \Rightarrow xyz\left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{\frac{1}{x}}&{\frac{1}{y}}&{\frac{1}{z}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{\frac{{xyz}}{x}}&{\frac{{xyz}}{y}}&{\frac{{xyz}}{z}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{yz}&{zx}&{xy}
\end{array}} \right| = {\text{R}}{\text{.H}}{\text{.S}}\]
Hence Proved.
Note: - In such types of questions first take \[x,y,z\] common from column 1, column 2, and column 3 respectively, then multiply $xyz$ in the third row of the determinant we will get the required result.
Consider L.H.S
\[ \Rightarrow \left| {\begin{array}{*{20}{c}}
{ax}&{by}&{cz} \\
{{x^2}}&{{y^2}}&{{z^2}} \\
1&1&1
\end{array}} \right|\]
Take \[x,y,z\] common from column 1, column 2, and column 3 respectively.
\[ \Rightarrow \left| {\begin{array}{*{20}{c}}
{ax}&{by}&{cz} \\
{{x^2}}&{{y^2}}&{{z^2}} \\
1&1&1
\end{array}} \right| = xyz\left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{\frac{1}{x}}&{\frac{1}{y}}&{\frac{1}{z}}
\end{array}} \right|\]
Now multiply $xyz$ in the third row of the determinant.
\[ \Rightarrow xyz\left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{\frac{1}{x}}&{\frac{1}{y}}&{\frac{1}{z}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{\frac{{xyz}}{x}}&{\frac{{xyz}}{y}}&{\frac{{xyz}}{z}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
a&b&c \\
x&y&z \\
{yz}&{zx}&{xy}
\end{array}} \right| = {\text{R}}{\text{.H}}{\text{.S}}\]
Hence Proved.
Note: - In such types of questions first take \[x,y,z\] common from column 1, column 2, and column 3 respectively, then multiply $xyz$ in the third row of the determinant we will get the required result.
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