If the angle made by the tangent at the point x 0 , y 0 on the curve x = 12 t + sin t cos t , y = 12 1 + sin t 2 , 0 < t < π 2 , with the positive x -axis is π 3 , then y 0 is equal to

Step-by-step solution

Given, Curve x = 12 t + sin t cos t y = 12 1 + sin t 2 Now d y d x = d y d t d x d t tan π 3 = d d t 12 1 + sin t 2 d d t 12 t + sin t cos t 3 = 12 × 2 1 + sin t cos t 12 1 + cos 2 t 3 = 2 1 + sin t · cos t 2 cos 2 t 3 = 1 + sin t cos t = cos t 2 + sin t 2 2 cos 2 t 2 - sin 2 t 2 3 = tan t 2 + π 4 π 3 = t 2 + π 4   ⇒ t 2 = π 12   ⇒   t = π 6 Now putting t = π 6 in y = 12 1 + sin t 2 as x 0 , y 0 lie on curve So y 0 = 12 1 + sin π 6 2 = 12