The distance from the origin, of the normal to the curve, x = 2 cos ⁡ t + 2 t sin ⁡ t , y = 2 sin ⁡ t - 2 t cos ⁡ t a t t = π 4 , is :

Step-by-step solution

At t = π 4 , x = 2 1 2 + 2 π 4 . 1 2 = 2 + π 2 2 = 4 + π 2 2 y = 2 1 2 - 2 π 4   ⋅   1 2   =   2 -   π 2 2 = 4 - π 2 2 d y d x = 2 cos ⁡ t - 2   cos ⁡ t + t   ( - sin ⁡ t ) = 2 t sin ⁡ t d x d t =   - 2 sin ⁡ t + 2   sin ⁡ t + t ⋅ cos ⁡ t = 2 t cos ⁡ t d y d x = tan ⁡ t at t = π 4   ⇒   d y d x = tan  ⁡ π 4 = 1 d y d x = 1 . Slope of t