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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Solution:Consider

a/b = c/d = e/f = k

So we get

a = bk, c = dk, e = fk

(i) LHS = (b

^{2}Â + d^{2}Â + f^{2}) (a^{2}Â + c^{2}Â + e^{2})We can write it as

= (b

^{2}Â + d^{2}Â + f^{2}) (b^{2}k^{2}Â + d^{2}k^{2}Â + f^{2}k^{2})Taking out the common terms

= (b

^{2}Â + d^{2}Â + f^{2}) k^{2}Â (b^{2}Â + d^{2}Â + f^{2})So we get

= k

^{2}Â (b^{2}Â + d^{2}Â + f^{2})RHS = (ab + cd + ef)

^{2}We can write it as

= (b. kb + dk. d + fk. f)

^{2}So we get

= (kb

^{2}Â + kd^{2}Â + kf^{2})Taking out common terms

= k

^{2}Â (b^{2}Â + d^{2}Â + f^{2})^{2}Therefore, LHS = RHS.