Question

For what value of x, the duplicate ratio of ` (x + a)/(x + b) " is " a/b` ?

Answer 1

Duplicate ratio of `(x +a)/(x + b) " is " ((x+a)^(2))/((x + b)^(2)) = (x^(2)+2xa + a^(2))/(x^(2) + 2 xb + b^(2))`
As per question, `(x^(2)+2 xa + a^(2))/(x^(2) + 2xb + b^(2)) = a/b`
or, ` ax^(2) + 2abx + ab^(2) = bx^(2) + 2abx + a^(2) b`
or, `ax^(2) - bx^(2) = a^(2) b - ab^(2)`
or, `x^(2) (a - b) = ab (a -b) " or, " x^(2) = (ab(a-b))/((a -b)) = ab`
` :. x^(2) = ab rArr x = pm sqrt(ab)`
` :. x = pm sqrt(ab)`.