Let $p, q$ and $r$ be the altitudes of a triangle with area $S$ and perimeter $2 t .$ Then, the value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$ is

• A

  $\frac{s}{t}$
• Correct answer
  B

  $\frac{t}{S}$
• C

  $\frac{s}{2 t}$
• D

  $\frac{2 S}{t}$

Solution

According to question, $S=\frac{1}{2} a \cdot p$ $$ \begin{array}{c} =\frac{1}{2} b \cdot q=\frac{1}{2} c \cdot r \\ \therefore \quad \frac{1}{p}=\frac{a}{2 S}, \frac{1}{q}=\frac{b}{2 S}, \frac{1}{r}=\frac{c}{2 S} \\ \therefore \quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{2 S}(a+b+c)=\frac{2 t}{2 S}=\frac{t}{S} \end{array} $$

Hence, the correct answer is option: 'B'.