Can someone show how to solve this problem

Question

Find the equation of the tangent line to the curve given by,

x=3e^t
y=5e^-t

at the point where t=0

Thanks!

Answer 1

one way:

when t = 0
x = 3
y = 5

dy/dx = dy/dt * dt/dx

dy/dt = -5 e^-t

dx/dt = 3 e^t
so
dt/dx = (1/3) e^-t

dy/dx = -5 e-t (1/3)e^-t
dy/dx = -(5/3) e^-2t = -5/3 when t = 0
so
y = -5/3 x + b
5 = -5/3 * 3 + b
b = 10
so
y = -5 x/3 + 10