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 need to solve the system of equations:
x = ln(sqrt(1+t^2) + t)
y = sqrt(t^2 + 1)
Let's begin by isolating t in the second equation:
y = sqrt(t^2 + 1)
Squaring both sides of the equation, we get:
y^2 = t^2 + 1
Rearranging the equation gives:
t^2 = y^2 - 1
Next, let's solve the first equation for t. Start by exponentiating both sides of the equation:
e^x = sqrt(1 + t^2) + t
Now, subtract t from both sides:
e^x - t = sqrt(1 + t^2)
Square both sides of the equation:
(e^x - t)^2 = 1 + t^2
Expanding the left side of the equation, we get:
e^2x - 2te^x + t^2 = 1 + t^2
Rearranging terms, we have:
e^2x - 2te^x - 1 = 0
Now, let's substitute t^2 with (y^2 - 1) in the above equation:
e^2x - 2te^x - 1 = 0
e^2x - 2tye^x - 1 = 0
Now, solve this equation for t:
2tye^x = e^2x - 1
t = (e^2x - 1) / (2ye^x)
Now that we have found the expression for t in terms of x and y, substitute this expression into the equation for y in terms of t:
y = sqrt(t^2 + 1)
y = sqrt[(e^2x - 1)^2 / (4y^2e^2x) + 1]
To simplify this expression, we can get rid of the square root by squaring both sides of the equation:
y^2 = (e^2x - 1)^2 / (4y^2e^2x) + 1
Now, multiply through by (4y^2e^2x) to eliminate the denominators:
4y^2e^2xy^2 = (e^2x - 1)^2 + 4y^2e^2x
Expand and simplify both sides of the equation:
4y^4e^2x = e^4x - 2e^2x + 1 + 4y^2e^2x
Rearranging terms:
4y^4e^2x - 4y^2e^2x + 1 = e^4x - 2e^2x
Factoring out common terms:
e^2x(4y^4 - 4y^2 + 1) = e^4x - 2e^2x
Divide through by e^2x:
4y^4 - 4y^2 + 1 = e^2x - 2
Now, rearrange terms to put it in the form y = f(x):
4y^4 - 4y^2 - e^2x + 1 = 0
This equation represents y as a function of x, eliminating the parameter t.