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Let a, b and c be the distinct non-negative numbers. If the vectors \(\rm \hat i + b\hat j + \hat k,\hat i + b\hat j, a\hat i + c^2\hat j + c\hat k\)lie on a plane, then which one of the following is correct1. c is the arithmetic mean of a and b 2. c is the geometric mean of a and b 3. c is the harmonic mean of a and b 4. c = 0 |
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Answer» Correct Answer - Option 2 : c is the geometric mean of a and b Concept: If G is the geometric mean of the numbers a and b and is given by ⇔ G = \(\rm \sqrt {ab}\)
Calculation: Given, \(\rm \hat i + b\hat j + \hat k,\hat i + b\hat j, a\hat i + c^2\hat j + c\hat k\) are coplanar ∴\(\rm \left | \begin{array}{ccc} 1 & b & 1 \\ 1 & b & 0 \\ a & c^2 & c \end{array} \right | =0\) ⇒ 1(bc) - b (c) + 1(c2 - ab) = 0 ⇒ c2 = ab ⇒ c = \(\rm \sqrt {ab}\) Hence, option (2) is correct. |
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