Find the corresponding Cartesian equation for each pair of parametric equations:
x = t - (1/t) and y = 2*[t + (1/t)]
The answer is y^2/16 - x^2/4 = 1
I have no idea where to begin. I tried squaring both equations in hope to easily remove t^2 but ended up with ugly numbers. Then I tried substituting the t value (which I rearranged from x) into the y equation but also ended up with ugly numbers.
Please show your working so I can understand how you came up with the answer. Thanks for your help.
(y/2)^2 = (t + 1/t)^2 = t^2+2+1/t^2
x^2 = (t - 1/t)^2 = t^2-2+1/t^2
y^2/4 - x^2 = 4
or in standard form,
y^2/16 - x^2/4 = 1
To find the corresponding Cartesian equation for a pair of given parametric equations, we need to eliminate the parameter (t) and express x and y in terms of each other.
Let's start by solving the first parametric equation, x = t - (1/t), for t.
Multiply both sides of the equation by t:
xt = t^2 - 1
Rearrange the equation:
t^2 - xt - 1 = 0
Now, apply the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -x, and c = -1. Plugging in these values, we get:
t = (x ± sqrt(x^2 + 4)) / 2
Now that we have an expression for t, let's substitute it into the second parametric equation, y = 2*[t + (1/t)].
y = 2 * [(x ± sqrt(x^2 + 4)) / 2 + (2/(x ± sqrt(x^2 + 4)))]
Now, simplify the equation:
y = (x ± sqrt(x^2 + 4)) + (4/(x ± sqrt(x^2 + 4)))
Combining like terms:
y = x ± sqrt(x^2 + 4) + (4/(x ± sqrt(x^2 + 4)))
To further simplify this equation, let's find a common denominator for the last two terms:
y = (x ± sqrt(x^2 + 4)) * (x ± sqrt(x^2 + 4)) / (x ± sqrt(x^2 + 4)) + (4/(x ± sqrt(x^2 + 4)))
Expanding the numerator:
y = (x^2 ± 2x*sqrt(x^2 + 4) + (x^2 + 4)) / (x ± sqrt(x^2 + 4)) + (4/(x ± sqrt(x^2 + 4)))
Simplifying:
y = (2x^2 + 4 ± 2x*sqrt(x^2 + 4)) / (x ± sqrt(x^2 + 4))
The resulting equation now only has x and y terms. Next, let's eliminate the square root by multiplying both sides by (x ± sqrt(x^2 + 4)):
y * (x ± sqrt(x^2 + 4)) = 2x^2 + 4 ± 2x * sqrt(x^2 + 4)
Expanding the left side:
yx ± y * sqrt(x^2 + 4) = 2x^2 + 4 ± 2x * sqrt(x^2 + 4)
Now, combine the terms with square roots on one side and the constant terms on the other side:
y * sqrt(x^2 + 4) ± 2x * sqrt(x^2 + 4) = 2x^2 - yx + 4
Factoring out the common factor of sqrt(x^2 + 4):
(sqrt(x^2 + 4))(y ± 2x) = 2x^2 - yx + 4
Finally, divide both sides by (y ± 2x):
(sqrt(x^2 + 4)) = (2x^2 - yx + 4) / (y ± 2x)
To simplify further, square both sides:
x^2 + 4 = [(2x^2 - yx + 4) / (y ± 2x)]^2
Expanding the numerator and denominator on the right side:
x^2 + 4 = (2x^2 - yx + 4)^2 / (y ± 2x)^2
Multiplying through by (y ± 2x)^2:
(x^2 + 4)(y ± 2x)^2 = (2x^2 - yx + 4)^2
Expanding both sides:
(x^2 + 4)(y^2 ± 4xy + 4x^2) = (2x^2 - yx + 4)(2x^2 - yx + 4)
Now, simplify both sides of the equation and combine like terms:
x^2y^2 ± 4x^3y + 4x^4 + 4y^2x^2 ± 16x^3 ± 16x^4 = 4x^4 - 4x^2xy + 16x^2 - 2xy(2x^2 - yx) + 16(2x^2 - yx) + y^2(2x^2 - yx) - 8(2x^2 - yx) + 16
Eliminating the common terms on both sides:
x^2y^2 ± 4x^3y + 4y^2x^2 ± 16x^3 = y^2(2x^2 - yx) − 8(2x^2 − yx)
Now, simplify and rearrange:
x^2y^2 ± 4x^3y + 4y^2x^2 ± 16x^3 = 2x^2y^2 − y^2x^2 − 16x^2 − 16yx + 8yx − 8y + 16
Combining like terms:
x^2y^2 − 2x^2y^2 ± 4x^3y + y^2x^2 − 16x^2x^2 + 16x^2 − 16xyx + 8yx − 4y + 16 − 16x^3 = 0
Simplifying further:
−x^2y^2 − 16x^4 + 16x^2y + 56x^2 + 4y − 16x^3 = 0
Finally, we can divide both sides of the equation by -16 and rearrange the terms:
x^2y^2/16 − x^4 − x^2y − 7/4x^2 − y/4 + x^3 = 0
Multiplying through by -4, we obtain:
-x^2y^2/4 + 4x^4 + 4x^2y + 7x^2 + y = 4x^3
Therefore, the corresponding Cartesian equation for the given pair of parametric equations is:
-x^2y^2/4 + 4x^4 + 4x^2y + 7x^2 + y = 4x^3