Question

The determinant $\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0,$ if

• A. x, y, z are in AP
• B. x, y, z are in GP
• C. x, y, z are in HP
• D. xy, yz, zx are in AP

Answer 1

The correct option is B x, y, z are in GP

Given, $\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$
Applying $C_1\to C_1-(p{C}_2+C_3)$
$\Rightarrow \left|\begin{array}{ccc}0& x& y\\ 0& y& z\\ -(x{p}^2+yp+yp+z)& xp+y& yp+z\end{array}\right|=0\Rightarrow -(x{p}^2+2yp+z)(xz-y^2)=0\therefore Eitherx{p}^2+2yp+x=0or{y}^2=xz$
$\Rightarrow$ x, y, z are in GP