Question

Let $a,b,c$ be distinct non-negative numbers. If the vectors $\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+b\hat{j}+b\hat{k}$ lie in a plane, then $c$ is

• A. A.M of $a$ and $b$
• B. G.M of $a$ and $b$
• C. H.M of $a$ and $b$
• D. None of these.

Answer 1

The correct option is D None of these.

For given planes to lie in a plane,

$\left|\begin{array}{ccc}1& a& c\\ 1& 0& 1\\ c& b& b\end{array}\right|=0$

$\Rightarrow -a(b-c)-b(1-c)=0$

$\Rightarrow ac+bc=ab+b\Rightarrow c=\frac{b(1+a)}{a+b}$