Eliminate the parameter to find the Cartesian Equation

Question

Eliminate the parameter to find the cartesian equation of the curve: x = 2sec⁡(theta), y = 2tan⁡(theta),−(pi)/2 < (theta) < (pi)/2.



The equation of the curve is: ?

Answer 1

If x = 2sec⁡(θ) then we can solve the θ:

x = 2sec⁡(θ)

x/2 = sec(θ)

θ = sec^-1(x/2)

Plugging "sec^-1(x/2)" in for "θ" in the other equation gives:

y = 2tan⁡(θ)

y = 2tan⁡(sec^-1(x/2))

Remembering that sec(θ) = hypotenuse/adjacent we can draw a triangle where "x" is the hypotenuse and "2" is the adjacent:

We can now use the Pythagorean Theorem to solve for the opposite side:

2² + (opposite)² = x²

(opposite)² = x² - 4

opposite = √(x² - 4)

Then, remembering that tan(θ) = opposite/adjacent:

tan(θ) = √(x² - 4)/2

So y = 2√(x² - 4)/2 or y = √(x² - 4)

Square both sides to get y² = x² - 4 then subtract x² from both sides to get:

y² - x² = -4 then divide both sides by -4 to get:

y²/-4 - x²/-4 = 1 or x²/4 - y²/4 = 1

Answer 2

x = 2secθ So, secθ = x/2

y = 2tanθ So, tanθ = y/2

Since 1 + tan²θ = sec²θ, we have 1 + y² / 4 = x² / 4. So, x² / 4 - y² / 4 = 1