For how many real values of λ do these three planes intersect in a common line?

$x + y + z = 1, x + 2 λ y + λ^{2} z = 3, x + y + λ z = 5.$

Question

For how many real values of λ do these three planes intersect in a common line? x + y + z = 1 , x + 2 λ y + λ 2 z = 3 , x + y + λ z = 5. ​

TestHub · Accepted Answer

For a common line, the normals must be dependent, but consistency must also be checked. At lambda=1 the first and third planes are distinct parallel planes. At lambda=1/2 the equations below are inconsistent. Therefore neither candidate works and the number is zero. $det M = (2 λ - 1) (λ - 1) = 0, λ = 1 or \frac{1}{2} . λ = 1 : x + y + z = 1, x + y + z = 5 (inconsistent). λ = \frac{1}{2} : x + y + z = 1, 2 x + 2 y + z = 10, 4 x + 4 y + z = 12. x + y = 9, z = - 8, 4 (x + y) + z = 28 \neq = 12. 0 .$

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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