Question

The determinant $\begin{vmatrix}xp+y&x&y\\ yp+z&y&z\\ 0&xp+y&yp+z\end{vmatrix} = 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, $\begin{vmatrix}x p+y & x & y \\ y p+z & y & z \\ 0 & x p+y & y p+z\end{vmatrix}=0$
Applying $C_{1} \rightarrow C_{1}-\left(p C_{2}+C_{3}\right)$
$\Rightarrow \begin{vmatrix}0 & x & y \\ 0 & y & z \\ -\left(x p^{2}+y p+y p+z\right) & x p+y & y p+z\end{vmatrix}=0$
$\Rightarrow -\left(x p^{2}+2 y p+z\right)\left(x z-y^{2}\right)=0$
$\therefore $ Either $x p^{2}+2 y p+z=0$ or $y^{2}=x z$
$\Rightarrow x, y, z$ are in GP.