Question

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

• A. $x,y,z$ are in A.P.
• B. $x,y,z$ are in G.P.
• C. $x,y,z$ are in H.P.
• D. $xy,yz,zx$ are in A.P.

Answer 1

The correct option is B $x,y,z$ are in G.P.

$\left|\begin{array}{ccc}xp+y& x& y\\ py+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$
Applying $C_1\to C_1-(p{C}_2+C_3)$
$\Rightarrow \left|\begin{array}{ccc}0& x& y\\ 0& y& z\\ -(x{p}^2+yp+yp+z)& xp+y& yp+z\end{array}\right|=0\Rightarrow -(x{p}^2+yp+yp+z)(xz-y^2)=0$
$\Rightarrow x{p}^2+2yp+z=0$ or $x,y,z$ in $G.P.$
Hence, option 'B' is correct.