A) A.P.
B) H.P.
C) G.P.
D) Arithmetico-geometric progression
Correct Answer: B
Solution :
Let \[{{P}_{1}},{{P}_{2}},{{P}_{3}}\] be altitudes from P, Q and R \[{{P}_{1}}=c\sin Q=\lambda bc\], \[{{P}_{2}}=a\,\sin R=\lambda ca\] \[{{P}_{3}}=b\sin P=\lambda ab\] \[\left[ \therefore \frac{\sin P}{a}=\frac{\sin Q}{b}=\frac{\sin R}{c}=\lambda \right]\] Þ \[{{P}_{1}},{{P}_{2}},{{P}_{3}}\] are in A.P. Þ \[\lambda bc,\,\,\lambda ca,\,\,\lambda ab\]are in A.P. Þ \[bc,\,\,ca,\,\,ab\] are in A.P. Þ \[\frac{abc}{a},\frac{abc}{b},\frac{abc}{c}\] are in A.P \[\frac{1}{a},\,\,\,\frac{1}{b},\,\,\,\frac{1}{c}\] are in A.P. \[\therefore a,\,\,b,\,\,c\] are in H.P. i.e., sides of the triangle are in H.P.You need to login to perform this action.
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