If (x square + y square) (a square+ b square) =(ax + by)^2. Prove that

1.	If (x square + y square) (a square+ b square) =(ax + by)^2. Prove that x/a = y/b
Answer» We have,(x^2+y^2)(a^2+b^2)=(ax+by)^2Prove that, \\dfrac{x}{a}=\\dfrac{y}{b}.∴ (x^2+y^2)(a^2+b^2)=(ax+by)^2⇒ x^2(a^2+b^2)+y^2(a^2+b^2)=(ax)^2+(by)^{2} +2(ax)(by)Using algebraic identity,(x+y)^{2}=x^{2}+y^{2}+2xy⇒ x^2a^2+x^2b^2+y^2a^2+y^2b^2=a^2x^2+b^2y^{2} +2abxy⇒ x^2b^2+y^2a^2=2abxy⇒ x^2b^2+y^2a^2-2abxy=0⇒ (xb)^2+(ya)^2-2(xb)(ya)=0Using algebraic identity,(x-y)^{2}=x^{2}+y^{2}-2xy⇒ (xb-ya)^2=0⇒ xb - ya = 0⇒ xb = ya⇒ \\dfrac{x}{a}=\\dfrac{y}{b}, proved.Hence, \\dfrac{x}{a}=\\dfrac{y}{b}, proved. (x²+y²) (a²+b²)= (ax+by)²(a²x²+b²x²+a²y²+b²y²)=(a²x²+b²x²+2abxy)b²x²+a²y²=2axbyb²x²+a²y²-2axby=0(bx-ay)²=0bx-ay=0Therefore, y/b=x/a

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