Question

Consider the triangle $ABC$ with usual notations and the following equations
$a^2{\sin }^{2}B+2a\sin B+|\sin \theta |=0$ ... (i)
$b^2{\sin }^{2}A+2b\sin A+|\sin \theta |=0$ ... (ii)
The number of values of $\theta$ in $[0,10\pi ]$ is

• A. $10$
• B. $4$
• C. $5$
• D. $15$

Answer 1

The correct option is A $10$

Clearly, $a\sin B$ and $b\sin A$, are the roots of the equation $x^2+2x+|\sin \theta |=0$
But $a\sin B=b\sin A$, hence roots are equal
$\Rightarrow$ Discriminant $=0$
$\Rightarrow 4-4|\sin \theta |=0$
$\Rightarrow |\sin \theta |=1$
Hence number of value of $\theta$ is $10$.