Question

(a2+b2) (x2+y2)=(ax+by)2;prove that a/x=b/y

Answer 1

refer pic for complete solution

Answer 2

$\dfrac{a}{x}=\dfrac{b}{y}$, proved.

Step-by-step explanation:

We have,

$(a^2+b^2)(x^2+y^2)=(ax+by)^2$

To prove that, $\dfrac{a}{x}=\dfrac{b}{y}$

∴ $(a^2+b^2)(x^2+y^2)=(ax+by)^2$

⇒ $a^2x^2+a^2y^2+b^2x^2+b^2y^2=(ax)^2+(by)^2+2(ax)(by)$

⇒ $a^2x^2+a^2y^2+b^2x^2+b^2y^2=a^2x^2+b^2y^2+2abxy$

Using the algebraic identity,

$(x+y)^{2}=x^{2}+y^{2}+2xy$

⇒ $a^2x^2+a^2y^2+b^2x^2+b^2y^2-(a^2x^2+b^2y^2+2abxy)=0$

⇒ $a^2x^2+a^2y^2+b^2x^2+b^2y^2-a^2x^2-b^2y^2-2abxy=0$

⇒ $a^2y^2+b^2x^2-2abxy=0$

⇒ $(ay)^2+(bx)^2-2(ay)(bx)=0$

⇒ $(ay-bx)^{2}=0$

⇒ $ay-bx=0$

⇒ ay = bx

⇒ $\dfrac{a}{x}=\dfrac{b}{y}$, proved.