Question

Consider three planes $2x+py+6z=8,x+2y+qz=5$ and $x+y+3z=4$. These planes intersect at a point if:

• A. $p=2,q\ne 3$
• B. $p\ne 2,q\ne 3$
• C. $p\ne 2,q=3$
• D. $p=2,q=3$

Answer 1

Answer: (B)

$p\ne 2,q\ne 3$

Let $A=\left|\begin{array}{ccc}2& p& 6\\ 1& 2& q\\ 1& 1& 3\end{array}\right|=2(6-q)-1(3p-6)+1(pq-12)$ (expanded along first row)
$=pq-3p-2q+6=(p-2)(q-3)$
Given set of planes will intersect at a point only if $A\ne 0\Rightarrow p\ne 2,q\ne 3$