Review question

# When can this equation involving algebraic fractions hold? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8407

## Question

In this question, $a$ and $b$ are distinct, non-zero real numbers, and $c$ is a real number.

1. Show that, if $a$ and $b$ are either both positive or both negative, then the equation $\frac{x}{x-a}+\frac{x}{x-b}=1$ has two distinct real solutions.

2. Show that the equation $\frac{x}{x-a}+\frac{x}{x-b}=1+c$ has exactly one real solution if $c^2=-\dfrac{4ab}{(a-b)^2}$. Show that this equation can be written $c^2=1-\left(\dfrac{a+b}{a-b}\right)^2$ and deduce that it can only hold if $0 < c^2 \leq 1$.