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 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

Answer: (B)

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