Assertion :If $cot A, cot B, cot C$ are in A.P, then $a^{2}, b^{2}, c^{2}$ are also in A.P. Reason: Three numbers $x, y, z$ are in A.P , if $2 y = x + z$ ,

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion false but Reason is true

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$cot A, cot B, cot C$ is in A.P gives
$2 cot B = cot A + cot C$
For $△ A B C$
Using
$\frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c} = k \Rightarrow s i n A = a k, s i n B = b k, s i n C = c k$

$c o s A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}, c o s B = \frac{c^{2} + a^{2} - b^{2}}{2 a c}, c o s C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$
We get
$\frac{2 (c^{2} + a^{2} - b^{2})}{2 a b c} = \frac{b^{2} + c^{2} - a^{2}}{2 a b c} + \frac{a^{2} + b^{2} - c^{2}}{2 a b c} \Rightarrow 2 b^{2} = a^{2} + c^{2}$
Hence $a^{2}, b^{2}, c^{2}$ is in A.P

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Assertion :If cotA,cotB,cotC are in A.P, then a2,b2,c2 are also in A.P. Reason: Three numbers x,y,z are in A.P , if 2y=x+z,

Assertion :If $cot A, cot B, cot C$ are in A.P, then $a^{2}, b^{2}, c^{2}$ are also in A.P. Reason: Three numbers $x, y, z$ are in A.P , if $2 y = x + z$ ,