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

$$\mathrm{If}\frac{x}{a}=\frac{y}{b}\mathrm{then}\mathrm{show}\mathrm{that}$$$$\frac{x^3+3{xy}^2}{a^3+3{ab}^2}=\frac{y^3+3x^{2}y}{b^3+3a^{2}b}.$$

Answered by deleteduser1 last updated on 23/Dec/23

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

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

$$\frac{x}{a}=\frac{y}{b}=k$$$$x=ak,y=bk$$$$lhs:$$$$\frac{{\left(ak\right)}^3+3\left(ak\right){\left(bk\right)}^2}{3{\left(ak\right)}^2\left(bk\right)+{\left(bk\right)}^3}=\frac{a^3{k}^3+3{ab}^2{k}^3}{3a^2{bk}^3+b^3{k}^3}$$$$=\frac{k^3(a^3+3{ab}^2)}{k^3(3a^{2}b+b^3)}=\frac{a^3+3{ab}^2}{3a^{2}b+b^3}=rhs$$

Answered by Frix last updated on 23/Dec/23

$$\frac{x(x^2+3y^2)}{a(a^2+3b^2)}=\frac{y(y^2+3x^2)}{b(b^2+3a^2)}$$$$\frac{x}{a}=\frac{y}{b}$$$$\frac{x^2+3y^2}{a^2+3b^2}=\frac{y^2+3x^2}{b^2+3a^2}$$$$\frac{b^2+3a^2}{a^2+3b^2}=\frac{y^2+3x^2}{x^2+3y^2}$$$$y=\frac{bx}{a}$$$$\frac{b^2+3a^2}{a^2+3b^2}=\frac{\frac{b^2{x}^2}{a^2}+3x^2}{x^2+\frac{3b^2{x}^2}{a^2}}\mathrm{true}$$

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