question_answer

If a triangle \[PQR\], \[\sin P,\ \sin Q,\ \sin R\]are in A.P., then [IIT 1998]

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.

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.