Question

If $sin\alpha ,sin\beta ,cos\alpha$ are in GP, then roots of equation $x^{2}sin\beta +2xcos\beta +sin\beta =0$ are

• A. Real
• B. imaginary
• C. depends on $\alpha ,\beta ,\gamma$
• D. data insufficient

Answer 1

The correct option is A

Real

$sin\alpha ,sin\beta ,cos\alpha inGP\Rightarrow si{n}^2\beta =sin\alpha cos\alpha \Rightarrow 2si{n}^2\beta =2sin\alpha cos\alpha \Rightarrow 2si{n}^2\beta =sin2\alpha \Rightarrow si{n}^2\beta =\frac{1}{2}sin2x\therefore 0\le si{n}^2\beta \le \frac{1}{2}\Rightarrow cose{c}^2\beta \ge 2$
Given equation
$x^{2}sin\beta +2xcos\beta +sin\beta =0\Rightarrow x^2+2xcot\beta +1=0\therefore \Delta =4co{t}^2\beta -4=4(co{t}^2\beta -1)=4(cose{c}^2\beta -2)=4(2-2)\ge 0$
$\therefore$ real roots