Question

If $x+y=a,x\times y=b$ and $x.a=1$, then find the correct ones:

• A. $x=\frac{a+a\times b}{a^2}$
• B. $y=\frac{(a^2-1)a-a\times b}{a^2}$
• C. $x=\frac{b+a\times b}{a^2}$
• D. $y=\frac{(b^2-1)b-a\times b}{a^2}$

Answer 1

Answer: (A), (B)

The correct options are
A $x=\frac{a+a\times b}{a^2}$
B $y=\frac{(a^2-1)a-a\times b}{a^2}$

Given $x+y=a$

$\Rightarrow y=a-x$ ...(1)

$x\times y=b$ ...(2)

$x.a=1$ ...(3)

From (1) and (2), we get

$x\times (a-x)=b\Rightarrow x\times a-x\times x=b\Rightarrow x\times a=b$

$\Rightarrow a\times (x\times a)=a\times b$

$\Rightarrow (a.a)x-(a.x)a=a\times b$

$\Rightarrow {\left|a\right|}^{2}x-1.a=a\times b$ (from (3))

$\Rightarrow x=\frac{a+a\times b}{a^2}$ and $y=\frac{(a^2-1)a-a\times b}{a^2}$