Question

Co-ordinates of a point on the curve y = x sin x at which the normal is parallel to the line 4x + 4y = 3 are ______.

Options

• `(- pi/2, - pi/2)`

• `(pi/2, - pi/2)`

• `(- pi/2, pi/2)`

• `(pi/2, pi/2)`

Answer 1

Co-ordinates of a point on the curve y = x sin x at which the normal is parallel to the line 4x + 4y = 3 are `underline((pi/2, pi/2))`.

Explanation:

y = x sin x    ...(i)

`therefore "dy"/"dx" = x cos x + sin x`

Slope of the normal = - `1/("dy"/"dx") = (- 1)/(x cos x + sin x)`

Slope of the given line is - 1.

Since the normal is parallel to the given line.

`therefore (- 1)/(x cos x + sin x) = - 1`

⇒ x cos x + sin x = 1

⇒ x = `pi/2`

From (i), y = `pi/2`

∴ Co-ordinates of the point are `(pi/2, pi/2)`.