Question

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 $2y=x+z$,

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion false but Reason is true

Answer 1

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 $△ABC$
Using

$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}=k\Rightarrow sinA=ak,sinB=bk,sinC=ck$

$cosA=\frac{b^2+c^2-a^2}{2bc},cosB=\frac{c^2+a^2-b^2}{2ac},cosC=\frac{a^2+b^2-c^2}{2ab}$

We get
$\frac{2(c^2+a^2-b^2)}{2abc}=\frac{b^2+c^2-a^2}{2abc}+\frac{a^2+b^2-c^2}{2abc}\Rightarrow 2b^2=a^2+c^2$

Hence $a^2,b^2,c^2$ is in A.P