The slope of the tangent to a curve C : y = y x at any point [ x , y ) on it is 2 e 2 x - 6 e - x + 9 2 + 9 e - 2 x . If C passes through the points 0 , 1 2 + π 2 2 and α , 1 2 e 2 α then e α is equal to

Step-by-step solution

Given, d y d x = 2 e 2 x - 6 e - x + 9 2 + 9 e - 2 x On rearranging we get, d y d x = e 2 x - 6 e x 2 e 2 x + 9 Integrating both side we get, y = e 2 x 2 - 2 tan - 1 2 e x 3 + c If the curve passes through the point 0 , 1 2 + π 2 2 Then c = 2 π 4 + tan - 1 2 3 So, curve will be y = e 2 x 2 - 2 tan - 1 2 e x 3 - π 4 - tan - 1 2 3 Again curve passes through the point α , 1 2 e 2 α Putting the value in curve equation we get, e 2 α 2 = e 2 α 2 - 2 tan - 1 2 e α 3 - π 4 - tan