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781.
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining $(0, \frac{3}{2}) and (\frac{1}{2}, 2)$ then

$|b - a| = 1$

$b = \frac{π}{2} + a$

$|a + b| = 1$

b = a

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$|b - a| = 1$

$y = x + \sin (y) \frac{dy}{dx} = \frac{1}{1 - \cos (y)} = \frac{\frac{1}{2} - 0}{2 - \frac{3}{2}} = 1 \Rightarrow \cos (y) = 0 \Rightarrow y = (2 n + 1) \frac{π}{2} Point lie on curve b = a + \sin (y) b - a = \sin (y) |b - a| = 1$

782.

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