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lf the equations \(2{x}-{k}{y}+5z=7\) and \(kx-8y-10z+14=0\) represents the same plane then \(k^{2}-k+1=\)

(A) \(4\)
(B) \(12\)
(C) \(21\)
(D) \(0\)


Answer: C
The given equations are \(2x-ky+5z=7\) and \(kx-8y-10z+14=0\).
If the two equations represent the same plane, the ratio of the coefficients should remain a constant.
\( \dfrac{2}{k} =\dfrac{-k}{-8} =\dfrac{5}{-10} =\dfrac{7}{-14} \)
i.e. \(k = -4\)
Substituting in the required expression,
\( {k}^{2} - k + 1 = 16 + 4 + 1 = 21\)
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