For what value of k will the following set of planes intersect in a line? x-2y-z = 0 x+9y-5z= 0 kx-y+z = 0

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to determine the value of k such that the three planes:

x - 2y - z = 0

x + 9y - 5z = 0

and kx - y + z = 0 intersect in a line.

If the three planes intersect in a line the rank of the coefficient matrix and the rank of the augmented matrix should be 2. Here the coefficient matrix and the the augmented matrix is the same. It is a matrix with rows, [1 -2 -1][1 9 -5][k -1 1]. The determinant of this matrix should be 0.

|1 -2 -1|

|1  9 -5| = 0

|k -1  1|

The determinant of the matrix is:

1*(9*1) - 1(-5*-1) + 2( 1*1) -2(-5*k) -1*(1*-1) + 1(9*k)

This should be equal to 0.

=> 1*(9*1) - 1(-5*-1) + 2( 1*1) -2(-5*k) -1*(1*-1) + 1(9*k) = 0

=> 9 - 5 + 2 + 10k + 1 + 9k = 0

=> 19k = -9 + 5 - 2 - 1

=> k = -7/19

The value of k for which the three matrices intersect in a line is k = -7/19

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