Question

In a $△ABC$, $r_1,r_2,r_3$ are in H.P if $a,b,c$ are in:

• A. A.P
• B. G.P
• C. H.P
• D. none of these

Answer 1

The correct option is A A.P

Given that $r_1,r_2,r_3$ are in $H.P$

Therefore, $\frac{1}{r_1},\frac{1}{r_2},\frac{1}{r_3}$ are in $A.P$

We know that $\Rightarrow r_1=\frac{\Delta }{s-a}$

$\Rightarrow \frac{1}{r_1}=\frac{s-a}{\Delta }$

$\Rightarrow \frac{1}{r_2}=\frac{s-b}{\Delta }$

$\Rightarrow \frac{1}{r_3}=\frac{s-c}{\Delta }$

$\Rightarrow \frac{s-a}{\Delta },\frac{s-b}{\Delta },\frac{s-c}{\Delta }$ are in $A.P$

$\Rightarrow (s-a),(s-b),(s-c)$ are in $A.P$

$\Rightarrow -a,-b,-c$ are in $A.P$

$\Rightarrow a,b,c$ are in $A.P$