If-x-a-y-b-then-show-that-x-3-3xy-2-a-3-3ab-2-y-3-3x-2-y-b-3-3a-2-b-

Question Number 202258 by MATHEMATICSAM last updated on 23/Dec/23

If \frac{x}{a} = \frac{y}{b} then show that

\frac{x^{3} + 3 {x y}^{2}}{a^{3} + 3 {a b}^{2}} = \frac{y^{3} + 3 x^{2} y}{b^{3} + 3 a^{2} b} .

Answered by AST last updated on 23/Dec/23

\frac{x (x^{2} + 3 y^{2}) = x^{3} (1 + \frac{3 y^{2}}{x^{2}})}{y^{3} (\frac{3 x^{2}}{y^{2}} + 1)} = \frac{a^{3}}{b^{3}} \times \frac{\frac{3 b^{2} + a^{2}}{a^{2}}}{\frac{3 a^{2} + b^{2}}{b^{2}}}

= \frac{a (3 b^{2} + a^{2})}{b (3 a^{2} + b^{2})} = \frac{a^{3} + 3 {a b}^{2}}{3 a^{2} b + b^{3}} .

Answered by Rasheed.Sindhi last updated on 23/Dec/23

\frac{x}{a} = \frac{y}{b} = k

x = a k, y = b k

l h s :

\frac{{(a k)}^{3} + 3 (a k) {(b k)}^{2}}{3 {(a k)}^{2} (b k) + {(b k)}^{3}} = \frac{a^{3} k^{3} + 3 {a b}^{2} k^{3}}{3 a^{2} {b k}^{3} + b^{3} k^{3}}

= \frac{k^{3} (a^{3} + 3 {a b}^{2})}{k^{3} (3 a^{2} b + b^{3})} = \frac{a^{3} + 3 {a b}^{2}}{3 a^{2} b + b^{3}} = r h s

Answered by Frix last updated on 23/Dec/23

\frac{x (x^{2} + 3 y^{2})}{a (a^{2} + 3 b^{2})} = \frac{y (y^{2} + 3 x^{2})}{b (b^{2} + 3 a^{2})}

\frac{x}{a} = \frac{y}{b}

\frac{x^{2} + 3 y^{2}}{a^{2} + 3 b^{2}} = \frac{y^{2} + 3 x^{2}}{b^{2} + 3 a^{2}}

\frac{b^{2} + 3 a^{2}}{a^{2} + 3 b^{2}} = \frac{y^{2} + 3 x^{2}}{x^{2} + 3 y^{2}}

y = \frac{b x}{a}

\frac{b^{2} + 3 a^{2}}{a^{2} + 3 b^{2}} = \frac{\frac{b^{2} x^{2}}{a^{2}} + 3 x^{2}}{x^{2} + \frac{3 b^{2} x^{2}}{a^{2}}} true

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