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.

$△=\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$
Now $R_3\to R_3-p{R}_1-R_2$
$\left|\begin{array}{ccc}px+y& x& y\\ py+z& y& z\\ -(x{p}^2+2py+z)& 0& 0\end{array}\right|$
$\Rightarrow (y^2-xz)(p^{2}x+2py+z)=0\Rightarrow y^2=xz$

x, y, z are in G.P.
So choice (B) is true.