For what value of k do the planes kx + 4y + z | vedclass

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(B) Three planes $a_1x + b_1y + c_1z = 0$,$a_2x + b_2y + c_2z = 0$,and $a_3x + b_3y + c_3z = 0$ intersect in a single line if the determinant of their

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$2$

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(B) Three planes $a_1x + b_1y + c_1z = 0$,$a_2x + b_2y + c_2z = 0$,and $a_3x + b_3y + c_3z = 0$ intersect in a single line if the determinant of their coefficients is zero,i.e.,$\begin{vmatrix} k & 4 & 1 \ 4 & k & 2 \ 2 & 2 & 1 \end{vmatrix} = 0$. Expanding the determinant along the first row: $k(k(1) - 2(2)) - 4(4(1) - 2(2)) + 1(4(2) - 2(k)) = 0$ $k(k - 4) - 4(4 - 4) + 1(8 - 2k) = 0$ $k^2 - 4k - 0 + 8 - 2k = 0$ $k^2 - 6k + 8 = 0$ Factoring the quadratic equation: $(k - 2)(k - 4) = 0$ So,$k = 2$ or $k = 4$. Checking the options,$k = 2$ is provided as option $B$.

For what value of $k$ do the planes $kx + 4y + z = 0$,$4x + ky + 2z = 0$,and $2x + 2y + z = 0$ intersect in a single line?

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