Q86 / 150 · MHT CET 2022 (07 Aug Shift 1) · Mathematics

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

  • A
    the arithmetic mean of a and b.
  • B
    the harmonic mean of a and $b$.
  • Correct answer
    C
    the geometric mean of $a$ and $b$.
  • D
    not arithmetic mean of a and b.

Solution

$\begin{aligned} & \left|\begin{array}{lll}a & a & c \\ 1 & 0 & 1 \\ c & c & b\end{array}\right|=0 \\ & \Rightarrow a b-c^2=0 \\ & \Rightarrow c^2=a b\end{aligned}$ $\Rightarrow$ The geometric mean of a and b is $\mathrm{c}$

Hence, the correct answer is option: 'C'.

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