Question

If $x:y$ be the sub-duplicate ratio of $(x-a):(y-a)$ , prove that $a=\frac{xy}{a+y}$

Answer 1

Solution:-

Given:- $x:y$ be the sub-duplicate ratio of $(x-a):(y-a)$

To prove:- $a=\frac{xy}{a+y}$

Proof:-

$x:y$ be the sub-duplicate ratio of $(x-a):(y-a)$, i.e.,

$x:y=\sqrt{x-a}:\sqrt{y-a}$

$\Rightarrow \frac{x}{y}=\frac{\sqrt{x-a}}{\sqrt{y-a}}$

Squaring both sides,

$\frac{x^2}{y^2}=\frac{x-a}{y-a}$

$\Rightarrow x^2(y-a)=y^2(x-a)\left\{\text{By cross multiplication}\right\}$

$\Rightarrow x^{2}y-x^{2}a=y^{2}x-y^{2}a$

$\Rightarrow x^{2}y-y^{2}x=x^{2}a-y^{2}a$

$\Rightarrow xy(x-y)=a(x^2-y^2)$

$\Rightarrow a=\frac{xy(x-y)}{(x-y)(x+y)}$

$\Rightarrow a=\frac{xy}{(x+y)}$