Lengths of the tangents from A,B and C to the incircle are in A.P., then (a) `r_(1), r_(2), r_(3)` are in H.P (b) `r_(1), r_(2), r_(3)` are in A.P (c)a, b, c are in A.P (d)`cos A = (4c -3b)/(2c)`

Options

A. `r_(1) , r_(2) r_(3)` are in H.P
B. `r_(1), r_(2), r_(3)` are in AP
C. a, b, c are in A.P
D. `cos A = (4c -3b)/(2c)`

Answer

A, C, D

Explanation

Given that `s -a, s-b, and s-c` are in A.P.
`rArr a, b ,c` are in A.P.
`rarr (Delta)/(s-a), (Delta)/(s-b),(Delta)/(s-c)` are in H.P
`rArr r_(1), r_(2), r_(3)` are in H.P.
Also, `cos A = (b^(2) + c^(2) -a^(2))/(2bc)`
`= (b^(2) + c^(2) - (2b -c)^(2))/(2bc) = (4c - 3b)/(2c)`