eliminate parameter t when
x= ln(sqrt(1+t^2)+t) and y = sqrt(t^2+1
so to produce an equation y=f(x).
To eliminate the parameter "t" and express "y" as a function of "x," we can start by eliminating "t" between the two given equations:
Starting with the first equation:
x = ln(sqrt(1+t^2) + t)
We can rewrite this equation as:
e^x = sqrt(1+t^2) + t
Now, let's square both sides of the equation to get rid of the square root:
e^(2x) = (1+t^2) + 2t*sqrt(1+t^2) + t^2
Simplifying, we have:
e^(2x) = 2t^2 + 2t*sqrt(1+t^2) + 1
Rearranging the terms, we get:
2t*sqrt(1+t^2) = e^(2x) - 2t^2 - 1
Now, let's square both sides again to eliminate the square root:
(2t*sqrt(1+t^2))^2 = (e^(2x) - 2t^2 - 1)^2
Expanding and simplifying the equation, we have:
4t^2*(1+t^2) = e^(4x) - 4t^2*e^(2x) + 4t^4 + 2e^(2x) - 4t^2 - 2e^(2x) + 1
Simplifying further:
4t^4 + 4t^2 + 1 = e^(4x) - 4t^2*e^(2x)
Rearranging terms one last time:
4t^4 + (1 - 4e^(2x))t^2 + (e^(4x) - 1) = 0
This equation is a quadratic equation in terms of "t^2." To express "y" as a function of "x," we can use the quadratic formula. Let's denote "A = 4," "B = (1 - 4e^(2x))," and "C = (e^(4x) - 1)." The quadratic formula states:
t^2 = (-B ± sqrt(B^2 - 4AC)) / (2A)
With the corresponding values, substituting into the formula:
t^2 = (-(1 - 4e^(2x)) ± sqrt((1 - 4e^(2x))^2 - 4 * 4 * (e^(4x) - 1))) / 8
Simplifying further:
t^2 = (-1 + 4e^(2x) ± sqrt(1 - 8e^(2x) + 16e^(4x) - 16e^(4x) + 16)) / 8
t^2 = (3 - 4e^(2x) ± 15**(1/2)) / 8
Now, let's consider the positive and negative square root separately.
For the positive square root:
t^2 = (3 - 4e^(2x) + sqrt(15)) / 8
And for the negative square root:
t^2 = (3 - 4e^(2x) - sqrt(15)) / 8
Since our objective is to express "y" as a function of "x," we can now substitute these values of "t^2" into the second given equation:
For the positive square root:
y = sqrt(t^2 + 1)
y = sqrt((3 - 4e^(2x) + sqrt(15)) / 8 + 1)
For the negative square root:
y = sqrt(t^2 + 1)
y = sqrt((3 - 4e^(2x) - sqrt(15)) / 8 + 1)
These two expressions represent "y" as a function of "x" after eliminating the parameter "t."