Question

If the system of equations $2x + 3y - z = 0, x + ky - 2z = 0$ and $2x - y + z = 0$ has a non-trivial solution $(x, y, z)$, then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k $ is equal to :

• A. $\frac{3}{4}$
• B. -4
• C. $\frac{1}{2}$
• D. $ - \frac{1}{4}$

Answer 1

Answer: (C)

$\begin{pmatrix}2&3&-1\\ 1&K&-2\\ 2&-1&1\end{pmatrix} = 0 $
By solving $ K = \frac{9}{2} $
$ 2x+3y-z=0 $ .....(1)
$ x+ \frac{9}{2} y-2z=0 $ ...(2)
$ 2x - y + z = 0 $ ....(3)
$ \left(1\right)-\left(3\right) \Rightarrow 4y - 2z = 0 $
$ 2y = z$ ...(4)
$ \frac{y} {z} = \frac{1}{2} $ ...(5)
put z from eqn. (4) into (1)
$ 2x + 3y - 2y = 0 $
$ 2x + y = 0 $
$ \frac{x}{y} = - \frac{1}{2} $ ....(6)
$ \frac{\left(6\right)}{\left(5\right)} \frac{z}{x} = - 4 $
$ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} + K = \frac{1}{2} $