Question

If $\alpha ,\beta ,\gamma$ are the altitudes of a $\Delta ABC$ and 2s denotes its perimeter then
${\alpha }^{-1}+{\beta }^{-1}+{\gamma }^{-1}$ is equal to

• A. $\frac{\Delta }{s}$
• B. $\frac{s}{\Delta }$
• C. $s.\Delta$
• D. none of these

Answer 1

The correct option is B $\frac{s}{\Delta }$

In $\Delta ABC$:
we know that, $△=\frac{a\alpha }{2}=\frac{b\beta }{2}=\frac{c\gamma }{2}$
now, ${\alpha }^{-1}+{\beta }^{-1}+{\gamma }^{-1}=\frac{a}{2△}+\frac{b}{2△}+\frac{c}{2△}=\frac{a+b+c}{2△}$
Therefore, ${\alpha }^{-1}+{\beta }^{-1}+{\gamma }^{-1}=\frac{2s}{2△}=\frac{s}{△}$
Ans: B