First slide

Theory of expressions

Question

Let $α, β, γ$ be distinct real numbers lying in $(0, π / 2)$ , then the equation $\frac{1}{x - \sin α} + \frac{1}{x - \sin β} + \frac{1}{x - \sin γ} = 0,$ has

Moderate

A

two distinct real roots
B

two equal roots
C

two imaginary roots
D

one real and one imaginary root.

Solution

Let $a = \sin α, b = \sin β, c = \sin γ$
Note that $a, b, c$ are distinct and $0 < a, b, c < 1 .$
We can write the given equation as
$\begin{matrix} f (x) = (x - b) (x - c) + (x - c) (x - a) \\ + (x - a) (x - b) = 0 \end{matrix}$
Assume that $a < b < c .$
Note that $f (a) = (a - b) (a - c) > 0$
$f (b) = (b - c) (b - a) < 0$
and $f (c) = (c - a) (c - b) > 0 .$
Thus, $f (x) = 0$ has a root in $(a, b)$ and a root in $(b, c)$ .

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