Consider the following.
x = e^t - 12
y = e^2t
(a) Eliminate the parameter to find a Cartesian equation of the curve.
from x = e^t - 12
e^t = x+12
t = ln(x+12)
form y = e^2t
2t = lny
t = (lny)/2
so (ln y)/2 = ln(x+12)
lny = 2ln(x+12)
lny = ln(x+12)^2
y = (x+12)^2
To eliminate the parameter t and find a Cartesian equation of the curve, we can solve the equations for t and substitute the value of t into the other equation.
Let's start by solving the second equation, y = e^2t, for t:
Take the natural logarithm (ln) of both sides to get:
ln(y) = ln(e^2t)
Using the property of logarithms (ln(e^x) = x), we can simplify this to:
ln(y) = 2t
Now, solve for t by dividing both sides by 2:
t = (1/2)ln(y)
Next, substitute this value of t into the first equation, x = e^t - 12:
x = e^(1/2 ln(y)) - 12
Using the property of logarithms (ln(e^x) = x), we can simplify this to:
x = sqrt(y) - 12
Therefore, the Cartesian equation of the curve is x = sqrt(y) - 12.