First slide

Theory of equations

Question

If $\sin θ, \sin α, \cos θ$ are in G.P., then the roots of $x^{2} + 2 x \cot α + 1 = 0$ are always

Moderate

A

equal
B

real
C

imaginary
D

greater than 1

Solution

It is given that $\sin θ, \sin α, \cos θ$ are in G.P.
$\begin{array}{l} ∴ \sin^{2} α = \sin θ \cos θ \\ \Rightarrow 2 \sin^{2} α = \sin 2 θ \Rightarrow 1 - \cos 2 α = \sin 2 θ (i) \end{array}$
Let $D$ be the discriminant of the equation $x^{2} + 2 x \cot α + 1 = 0 .$
Then,
$\begin{array}{l} D = 4 \cot^{2} α - 4 = \frac{4 \cos 2 α}{\sin^{2} α} = \frac{4 (1 - \sin 2 θ)}{\sin^{2} α} [Using (i)] \\ \Rightarrow D = 4 {(\frac{\cos θ - \sin θ}{\sin α})}^{2} \geq 0 \end{array}$
Hence, the roots of the given equation are real.

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