Matrices and Determinants 2 Question 18
20. The determinant $\begin{vmatrix}x p+y & x & y \\ y p+z & y & z \\ 0 & x p+y & y p+z\end{vmatrix}=0$, if
(a) $x, y, z$ are in $\mathrm{AP}$
(b) $x, y, z$ are in GP
(c) $x, y, z$ are in HP
(d) $x y, y z, z x$ are in AP
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Answer:
Correct Answer: 20. (b)
Solution:
- Given, $\begin{vmatrix}x p+y & x & y \\ y p+z & y & z \\ 0 & x p+y & y p+z\end{vmatrix}=0$
Applying $C_{1} \rightarrow C_{1}-\left(p C_{2}+C_{3}\right)$
$ \begin{aligned} & \Rightarrow \quad\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 \quad-\left(x p^{2}+2 y p+z\right)\left(x z-y^{2}\right)=0 \end{aligned} $
$ \therefore \quad \text { Either } x p^{2}+2 y p+z=0 \text { or } y^{2}=x z $
$\Rightarrow x, y, z$ are in GP.
