Given the parametric equations below eliminate the parameter

Given the parametric equations below, eliminate the parameter t to obtain an equation for y as a function of x {x(t) = e^5t y(t) = e^2t y(x) = Given the parametric equations below, eliminate the parameter t to obtain an equation for y as a function of x {x(t) = 5 squareroot t y(t) = 2t + 8 y(x) =

Solution

1) x(t) = e^5t ;

Take natural log on both sides:

lnx = 5t ----(1)

t = (1/5)ln(x) = ln(x^1/5)

y(t) = e^2t

Take natural log on both sides:

lny = 2t ---- (2)

Divide equ (2) by equ (1):

lny/lnx = 2/5

5lny = 2lnx ----> lny^5 = lnx^2

y^5 = x^2

2) x(t) = 5sqrt(t)

square on both sides:

x^2 = 25t ----(1)

y(t) = 2t +8 ----(2)

Substitute equ value of t in equation in 2:

y -8 = 2t

y -8 = 2x^2/25

25(y -8) = 2x^2