Question

(a) Give a set of parametric equations (with domain) for the line segment from (4, -1) to (5,6). (b) Give a set of parametric equations (with domain) for the ellipse centered at (0,0) passing through the points (4,0), (-4,0), (0,3), and (0, -3), traversed once counter-clockwise. (c) Find the (x, y) coordinates of the points where the curve, defined parametrically by I= 2 cost y = sin 2t 0<t<T, has a horizontal tangent.

Answer 1

@ The given line segment from (1, - 2) to (5,6). ie 2:1-5 6 y : -1 Taking parametrie equation given below: x = 4 - (4t-5t) = 9+t ost <1 y = -1-(-+-60) -1+7 2 Set of parametric equation is X29+ t, 42-1+ 7+ <t31 6 given ellipse centered at (0,0) and passing through the points (4,0), (-1,0), (0, 3), (0,-3).counterclockwise. qy so the equation of the ellipse is (0,3) Y 2 + 2 1 72 32 (0,0) >x le 2 16 + solo 31 (4,0) (0,-3) cit Taking parametric equation as ; x= 9 Cost ost <22 y = 3 sint todos

© Given parametric curve is (1 X z 2 cost ostat } y 2 Sin 2t dim dx dt =-2 Sint dy dt 22cos 2t > Cos at dy da 2COS 2t dy dt 2 dt dx Sint -2 Sint If the curve has a horizontal a horizontal tangent then COS2t = 0 i e 2t - 1 2t 2 37 31 2 ie tz 7/7 t 23 3 When t2 T . 4 2 = 2 x = 2los À 2 The point (264) is -15% alled [ qz sin T 2 1 so the point is (ar 1) ie (12, 1) 1 when tz 37 1 . 2 2.(-2) 37 x22cos 2 z 2 The point (2,4) is ya 2 Sin 3 -1 2 so the point is (-1) ie (-2) ie (-12,-1)