Question

If m : n is the duplicate ratio of m + x : n + x; then $x^2=mn$.

• A. True
• B. False

Answer 1

The correct option is A

True

Duplicate ratio of $a:b=(a\times a):(b\times b)$

$=a^2:b^2$

$\therefore (m+x)(m+x):(n+x)(n+x)=m:n$

$\frac{(m+x{)}^2}{(n+x{)}^2}=\frac{m}{n}$

$n(m^2+2mx+x^2)=m(n^2+2nx+x^2)$ $\left(\right(a+b{)}^2=a^2+b^2+2ab)$

$\Rightarrow n{m}^2+2mnx+n{x}^2=m{n}^2+2mnx+m{x}^2$

$\Rightarrow m^{2}n-m{n}^2+n{x}^2-m{x}^2=0$

$=mn(m-n)-x^2(m-n)=0$

$mn(m-n)=x^2(m-n)$

$=x^2=mn\frac{(m-n)}{(m-n)}=mn$