This is an important ques from the Book ML Aggarwal class 10th,, chapter – 7, ratio and proportion.
It is given that a/b = c/d = e/f
And we hve to prove that: (i) (b2 + d2 + f2) (a2 + c2 + e2) = (ab + cd + ef)2
Question 17, exercise 7.2
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If a/b = c/d = e/f prove that: (i) (b2 + d2 + f2) (a2 + c2 + e2) = (ab + cd + ef)2
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Solution:
Consider
a/b = c/d = e/f = k
So we get
a = bk, c = dk, e = fk
(i) LHS = (b2 + d2 + f2) (a2 + c2 + e2)
We can write it as
= (b2 + d2 + f2) (b2k2 + d2k2 + f2k2)
Taking out the common terms
= (b2 + d2 + f2) k2 (b2 + d2 + f2)
So we get
= k2 (b2 + d2 + f2)
RHS = (ab + cd + ef)2
We can write it as
= (b. kb + dk. d + fk. f)2
So we get
= (kb2 + kd2 + kf2)
Taking out common terms
= k2 (b2 + d2 + f2)2
Therefore, LHS = RHS.