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

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

Given that,
$\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$
Operating $C_1\to C_1-p{C}_2-C_3$, we get
$\left|\begin{array}{ccc}0& x& y\\ 0& y& z\\ -(x{p}^2+2py+z)& xp+y& yp+z\end{array}\right|=0or,(xz-y^2)(x{p}^2+2py+z)=0or,xz-y^2=0or,y^2=xzx,y,z\text{are in G.P.}$