Triangles And Properties Of Triangle Question 40
Question: Let $ d_1,d_2 $ and $ d_3 $ be the lengths of perpendiculars from circumventer of $ \Delta ABC $ on the sides BC, AC and AB, respectively, if $ \lambda ( \frac{a}{d_1}+\frac{b}{d_2}+\frac{c}{d_3} )=\frac{abc}{d_1d_2d_3} $ then $ \lambda $ equals
Options:
A) 1
B) 2
C) 3
D) 4
Show Answer
Answer:
Correct Answer: D
Solution:
[d] We have $ \tan A=\frac{a}{2d_1}; $ $ d_1=R\cos A $ etc. Similarly $ \tan B=\frac{b}{2d_2} $ and $ \tan C=\frac{C}{2d_3} $ In $ \Delta ABC,\tan A+\tan B+\tan C $ $ =\tan A\cdot \tan B\cdot \tan C $
$ \Rightarrow \frac{a}{2d_1}+\frac{b}{2d_2}+\frac{c}{2d_3}=\frac{abc}{8d_1d_2d_3} $
$ \therefore ,4( \frac{a}{d_1}+\frac{b}{d_2}+\frac{c}{d_3} )=\frac{abc}{d_1d_2d_3} $
$ \Rightarrow \lambda =4 $
BETA
