In ABC- if A - B - C be in A- P- then a 2- b 2- c 2 are inA- G-P-B- H-P-C- A-P-D- None of these

The correct option is C A.P. cot A, cot B and cot C are in A.P. ⇒ cot A + cot C = 2 cot B ⇒cos A/sin A + cos C/sin C = 2 cos B/sin B ⇒b^2 + c^2 a^2/2bc(ka ...

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Standard XII

Mathematics

Cosine Rule

In ABC, if A ...

Question

In $△$ ABC, if cot A, cot B, cot C be in A. P. then $a^{2}, b^{2}, c^{2}$ are in

H.P.

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G.P.

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A.P.

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None of these

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Solution

The correct option is C
A.P.

cot A, cot B and cot C are in A.P.
$\Rightarrow$ cot A + cot C = 2 cot B $\Rightarrow \frac{c o s A}{s i n A} + \frac{c o s C}{s i n C} = \frac{2 c o s B}{s i n B}$
$\Rightarrow \frac{b^{2} + c^{2} - a^{2}}{2 b c (k a)} + \frac{a^{2} + b^{2} - c^{2}}{2 a b (k c)} = 2 \frac{a^{2} + c^{2} - b^{2}}{2 a c (k b)}$
$\Rightarrow a^{2} + c^{2} = 2 b^{2} . Hence a^{2}, b^{2}, c^{2} are in A. P.$

It might help to remember this result as a general relation.

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In △ ABC, if cot A, cot B, cot C be in A. P. then a2,b2,c2 are in

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