Question

If $\cot A+\cot B+\cot C=k(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2})$, then the value of $k$ is

• A. $R^2$
• B. $rR$
• C. $\Delta$
• D. $a^2+b^2+c^2$

Answer 1

Answer: (C)

$\Delta$

$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{a^2}{4{△}^2}+\frac{b^2}{4{△}^2}+\frac{c^2}{4{△}^2}=\frac{a^2+b^2+c^2}{4{△}^2}$

and $\cot A+\cot B+\cot C=\frac{R}{abc}$

$(b^2+c^2-a^2+c^2+a^2-b^2+a^2+b^2-c^2)$

$=\frac{R}{abc}(a^2+b^2+c^2)=\frac{R}{abc}(\frac{4{△}^2}{x^2}+\frac{4{△}^2}{y^2}+\frac{4{△}^2}{z^2})$

$=\frac{4{△}^{2}R}{abc}(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2})$

$=\frac{4△R}{abc}.△(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2})=△(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2})$

$\therefore k=△$