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If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining $\left(0,{3\over2}\right)$ and $\left({1\over2},2\right)$, then

(1) |b – a| = 1

(2) $b={\pi\over2}+a$

(3) |a + b| = 1

(4) b = a
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Ans. (1) |b – a| = 1

Sol. y = x + siny

${dy\over dx}={1\over1-\cos y}={{1\over2}-0\over2-{3\over2}}=1$

$\implies$ cos y = 0

$\implies y = (2n + 1){\pi\over2}$

Point lie on curve b = a + sin y

b – a = sin y

|b – a| = 1

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