Question

100 Points! Help Please! I would be extremely grateful!

Answer 1

Answer should be - b/a cot t

Answer 2

x = acost

y = bsint

method 1 :-

x = acost

differentiate wrt t

dx/dt = a. d(cost)/dt = - asint ----(1)

y = bsint

differentiate wrt t
dy/dt = b.d(sint)/dt = bcost -----(2)

divide (2) ÷ (1)

(dy/dt )/(dx/dt) = (bcost)/(-asint)
dy/dx = -b/a { cost/sint}

dy/dx = -(b/a)cot t




method 2 :-

x = acost

x/a = cost
take square both sides

x²/a² = cos²t ---(1)

similarly ,

y²/b² = sin²t ----(2)

we know ,
sin²t + cos² t = 1 { trigo identity

put eqns (1) and (2)

x²/a² + y²/b² = 1
now, differentiate wrt x

2x/a² + 2y/b²dy/dx = 0

x/a² = -y/b²dy/dx

dy/dx = - (x/y)(b²/a² )

put x = acost and y = bsint

dy/dx = -(acost/bsint)(b²/a²) = -(b/a )cot t