First slide

Theory of equations

Question

If a, b, c are distinct positive numbers, then the nature of roots of the equation $\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = \frac{1}{x}$ is

Moderate

A

all real and distinct
B

all real and at least two are distinct
C

at least two real
D

all non-real

Solution

The equation on simplifying gives
$x (x - b) (x - c) + x (x - c) (x - a) + x (x - a) (x - b) - (x - a) (x - b) (x - c) = 0$ (1)
Let $f (x) = x (x - b) (x - c) + x (x - c) (x - a) + x (x - a) (x - b) - (x - a) (x - b) (x - c)$
We can assume without loss of generality that a < b < c. Now,
$\begin{array}{l} f (a) = a (a - b) (a - c) > 0 \\ f (b) = b (b - c) (b - a) < 0 \\ f (c) = c (c - a) (c - b) > 0 \end{array}$
So, one root of (1) lies in(a, b) and one root in (b, c). Obviously the third root must also be real.

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