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Conditions for a system of linear equations to have infinite solutions

lf the equati...

Question

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

4

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12

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21

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0

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Solution

The correct option is D $21$
The given equations are $2 x - k y + 5 z = 7$ and $k x - 8 y - 10 z + 14 = 0$ .
If the two equations represent the same plane, the ratio of the coefficients should remain a constant.
$\frac{2}{k} = \frac{- k}{- 8} = \frac{5}{- 10} = \frac{7}{- 14}$
i.e. $k = - 4$
Substituting in the required expression,
$k^{2} - k + 1 = 16 + 4 + 1 = 21$

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