Question

Find the Cartesian equation of the curve with parametric equations x = 2 sec(t) and y = 3 tan(t) for t ∈ π/2 , 3π/2, and describe the curve.

Answer 1

4y^2 - 9x^2 = -36

Explanation:

x = 2 sec t

y = 3 tan t

x^2 = 4 sec^2 t

y^2 = 9 tan^2 t

Now sec^2t = 1 + tan^2 t so we have:

4(1 + tan^2 t) + 9 tan^2 t = x^2 + y^2

4 + 13 tan^2 t = x^2 + y^2

13 tan^2 t = x^2 + y^2/ - 4

tan^2 t = x^2/13 + y^2/13 - 4/13

But , from the second equation tan t = y/3 so tan^2 t = y^2/9, so:

y^2/9 = x^2/13 + y^2/13 - 4/13

LCM of 9 and 13 is 117 so multiply thru by 117:

13y^2 = 9y^2 + 9x^2- 36

4y^2 - 9x^2 = -36