question_answer

If in a \[\Delta \text{ABC}\], the altitudes from the vertices A, B and C on opposite sides are in HP, then sin A, sin B, sin C are in

A) HP

B) arithoietic-geometric progression

C) AP

D) GP

Correct Answer: C

Solution :

In \[\Delta BDA,\]\[\cos ({{90}^{o}}-B)=\frac{AD}{c}\] \[\Rightarrow \]               \[AD=c\sin B\] Similarly, \[BE=a\sin C\]and \[CF=b\sin A\] Since, \[AD,BE,CF\]are in HP. \[\therefore \]  \[c\sin B,a\sin C,b\sin A\]are in HP. \[\Rightarrow \frac{1}{\sin C\sin B},\frac{1}{\sin A\sin C},\frac{1}{\sin B\sin A}\]are in AP \[\Rightarrow \] \[\sin A,\sin B,\sin C\]are in AP. (multiply by \[\text{sinA sinB sinC}\])