If \frac{x}{a}=\frac{y}{b}=\frac{z}{c} show that \frac{x^3}{a^3}+\frac{y^3}{b^3}+\frac{z^3}{c^3}=\frac{3 x y z}{a b c}.

ICSE Class X Math 2016 Solved Paper - Question 33

Question : 33 of 57

Marks: +1, -0

\;{x}/{a}=\;{y}/{b}=\;{z}/{c}

show that

\;{x^3}/{a^3}+\;{y^3}/{b^3}+\;{z^3}/{c^3}=\;{3 x y z}/{a b c}

Solution:

Given:

\;\; \;{x}/{a}=\;{y}/{b}=\;{z}/{c}=k

(say)

\;⇒ \;\; x=a k, y=b k, z=c k

\;\;\text " L.H.S. "\;= \;{x^3}/{a^3}+\;{y^3}/{b^3}+\;{z^3}/{c^3}

\;\; = \;{a^3 k^3}/{a^3}+\;{b^3 k^3}/{b^3}+\;{c^3 k^3}/{c^3}

\;\; = k^3+k^3+k^3=3 k^3

\;\; = 3 k ⋅ k ⋅ k

\;\; = 3 \;{x}/{a} \;{y}/{b} \;{z}/{c}

( ∵ x=a k, y=b k, z=c k)

\;\; = \;{3 x y z}/{a b c}=\;\text " R.H.S. "\;

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