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Question

If four points P (1,2,3) , Q (2,3,4), R(3,4,5), S(4,5, k) are coplanar then the value of k is / are -


A

6

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B

4

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C

5

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D

Any real number.

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Solution

The correct option is D

Any real number.


We have seen that if four points A(x1,y1,z1), B(x2,y2,z2), C(x3,y3,z3) and D(x4,y4,z4) are coplanar then -

∣ ∣x2x1y2y1z2z1x3x1y3y1z3z1x4x1y4y1z4z1∣ ∣=0

We’ll do the appropriate substitution -

∣ ∣2132433142534152k3∣ ∣=0Or∣ ∣11122233k3∣ ∣=0

Since first and second column are same, this determinant value is zero immaterial of k. k can be any real number.


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