Question

If $p_1,p_2,p_3$ are altitudes of a triangle $ABC$ from the vertices A, B, C and $\Delta$ is the area of the triangle, then ${p_1}^{-2}+{p_2}^{-2}+{p_3}^{-2}$ is equal to

• A. $\frac{a+b+c}{\Delta }$
• B. $\frac{a^2+b^2+c^2}{4{\Delta }^2}$
• C. $\frac{a^2+b^2+c^2}{{\Delta }^2}$
• D. none of these

Answer 1

Answer: (B)

$\frac{a^2+b^2+c^2}{4{\Delta }^2}$

Clearly, $p_1=\frac{2\Delta }{a}$
$\therefore p_1^{-2}+p_{-2}^2+p_3^{-2}=\frac{a^2}{4{\Delta }^2}+\frac{b^2}{4{\Delta }^2}+\frac{c^2}{4{\Delta }^2}$
$\therefore B$ is correct