If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining $\left(0, \frac{3}{2}\right) \mathrm{and} \left(\frac{1}{2}, 2\right)$ then

• $\left|\mathrm{b} - \mathrm{a}\right| = 1$

• $\mathrm{b} = \frac{\mathrm{\pi }}{2} + \mathrm{a}$

• $\left|\mathrm{a} +\mathrm{b}\right| = 1$

• b = a

$\mathrm{The} \mathrm{derivative} \mathrm{of} {\tan }^{-1}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - 1}{\mathrm{x}}\right) \mathrm{with} \mathrm{respect} \mathrm{to} {\tan }^{-1}\left(\frac{2\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}}{1 - 2{\mathrm{x}}^2}\right) \mathrm{at} \mathrm{x} = \frac{1}{2} \mathrm{is} :$

• $\frac{\sqrt{3}}{10}$

• $\frac{\sqrt{3}}{12}$

• $\frac{2\sqrt{3}}{5}$

• $\frac{2\sqrt{3}}{3}$