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If the system of equations, 2x + 3y-z = ...

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 `(x)/(y) + (y)/(z) + (z)/(x) + k` is equal to

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The correct Answer is:
2
`D = |{:(2,3,-1),(1,k,-2),(2,-1,1):}| = 0, 2 (k - 2) - 3 xx 5 + 2k + 1 = 0`
4k = 10, `k = (9)/(2)`
Equation (i) `2. (x)/(y) + 3 - (z)/(y) = 0`
Equation (ii) `2. (x)/(y) - 1 + (z)/(y) = 0`
An adding `(X)/(y) = (-1)/(2)`
`(z)/(y) = 2`
`(x)/(z) = - (1)/(4)`
`(x)/(y) + (x)/(z) + (z)/(y) + k = (1)/(2)`
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