Question
If in a $$\Delta \,ABC,$$ the altitudes from the vertices $$A, B, C$$ on opposite sides are in H.P, then $$\sin A, \sin B, \sin C$$ are in
A.
G.P.
B.
A.P.
C.
A.P. - G.P.
D.
H.P.
Answer :
A.P.
Solution :
$$\eqalign{
& \Delta = \frac{1}{2}{p_1}a = \frac{1}{2}{p_2}b = \frac{1}{2}{p_3}b \cr
& {p_1},{p_2},{p_3}\,\,{\text{are in H}}{\text{.P}}{\text{.}} \cr
& \Rightarrow \,\,\frac{{2\Delta }}{a},\frac{{2\Delta }}{b},\frac{{2\Delta }}{c}\,\,{\text{are in H}}{\text{.P}}{\text{.}} \cr
& \Rightarrow \,\,\frac{1}{a},\frac{1}{b},\frac{1}{c}\,\,{\text{are in H}}{\text{.P}}{\text{.}} \cr
& \Rightarrow \,\,a,b,c\,\,{\text{are in A}}{\text{.P}}{\text{.}} \cr} $$
$$ \Rightarrow \,\,K\sin A,K\sin B,K\sin C$$ are in A.P.
$$ \Rightarrow \,\,\sin A,\sin B,\sin C\,\,{\text{are in A}}{\text{.P}}{\text{.}}$$