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
• B. 4
• C. 5
• D. Any real number.

Answer 1

The correct option is D

Any real number.

We have seen that if four points $A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3)andD(x_4,y_4,z_4)$ are coplanar then -

$\left|\begin{array}{ccc}x2-x1& y2-y1& z2-z1\\ x3-x1& y3-y1& z3-z1\\ x4-x1& y4-y1& z4-z1\end{array}\right|=0$

We’ll do the appropriate substitution -

$\left|\begin{array}{ccc}2-1& 3-2& 4-3\\ 3-1& 4-2& 5-3\\ 4-1& 5-2& k-3\end{array}\right|=0Or\left|\begin{array}{ccc}1& 1& 1\\ 2& 2& 2\\ 3& 3& k-3\end{array}\right|=0$

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