Question

If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.

Options

• b = `π/2 + a`

• |a + b| = 1

• |b – a| = 1

• b = a

Answer 1

If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then |b – a| = 1.

Explanation:

Given y = x + siny at a point (a, b)   ...(i)

parallel to `(0, 3/2)(1/2, 2)`

`(dy)/(dx) = 1 + cosy (dy)/(dx)`

`(dy)/(dx) = 1/(1 - cosy)`

`(dy)/(dx)(a, b) = 1/(1 - cosb)`  ...(i)

Slope `(2 - 3/2)/(1/2 - 0) = 1/(1 - cosb)`  ...(ii)

Now, `1/(1 - cosb)` = 1

cosb = 0

b = `π/2, (3π)/2`

Equation (i)

b = a + sinb

b = a + 1  ...`("For"  b = π/2)`

b = a – 1   ...`("For"  b = (3π)/2)`

|b – a| = 1