Q68 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

If the reciprocals of the lengths of the sides of a $\triangle A B C$ are in harmonic progression, then its ex-radii $r_1, r_2, r_3$ are in

  • A
    Arithmetic progression
  • B
    Geometric progression
  • Correct answer
    C
    Harmonic progression
  • D
    Arithmetico-geometric progression

Solution

Let the sides of a $\triangle A B C$ are $a, b$ and $c$ $\because \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in H.P. $\Rightarrow a, b, c$ are in A.P. $\Rightarrow s-a, s-b, s-c$ are in A.P. $\Rightarrow \frac{s-a}{\Delta}, \frac{s-b}{\Delta}, \frac{s-c}{\Delta}$ are in A.P. $\therefore r_1, r_2, r_3$ are in H.P.

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

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