Question

In $△ABC$, If $x,y,z$ are perpendicular distance from the circumcentre to the sides of the $△ABC$ respectively. then $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}$

• A. $\frac{abc}{4xyz}$
• B. $\frac{4abc}{xyz}$
• C. $\frac{xyz}{4abc}$
• D. none of these

Answer 1

The correct option is D none of these

let $a=2R\sin Ab=2R\sin B$,$c=RsinC$

and also

$x=2R\cos Ay=2R\cos B,z=2R\cos C$

$\therefore \frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\tan A+\tan B+\tan C$

$\frac{a\cdot bc}{xyz}=\tan A\cdot \tan B\cdot \tan C$

In Triangle $ABC$ using property

$\sum \tan A=\pi \tan A$

Hence from above equations

$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{abc}{xyz}$

option $D$ is correct answer