Trigonometric Equations Question 168
Question: If a triangle $ PQR $ , $ \sin P,\ \sin Q,\ \sin R $ are in A.P., then
[IIT 1998]
Options:
A) The altitudes are in A.P.
B) The altitudes are in H.P.
C) The medians are in G.P.
D) The medians are in A.P.
Show Answer
Answer:
Correct Answer: B
Solution:
- $ PB=QC=l $ are in A.P.
$ \Rightarrow $ $ a,,b,,c $ are in A.P.
$ \therefore $ $ \frac{\sin P}{a}=\frac{\sin Q}{b}=\frac{\sin R}{c}=\lambda $ Let $ p_1,,p_2,,p_3 $ be altitudes from $ P,,Q,,R $ $ p_1=c\sin Q=\lambda bc $ , $ p_2=a\sin R=\lambda ac, $ $ p_3=b\sin P=\lambda ab $ Since $ a,,b,,c $ are in A.P. Hence $ \frac{1}{a},,\frac{1}{b},,\frac{1}{c} $ are in H.P.
$ \Rightarrow $ $ \frac{abc}{a},,\frac{abc}{b},,\frac{abc}{c} $ are in H.P. $ \Rightarrow $ $ bc,,ac,,ab $ are in H.P.
$ \Rightarrow $ $ \lambda bc,,\lambda ac,,\lambda ab $ are in H.P.
$ \Rightarrow $ $ p_1,,p_2,,p_3 $ are in H.P.
BETA
