Can You Convert a Cartesian Equation to Parametric Form?

[Moore1879]

Okay, is it possible to transform an "x-y" equation into a parametric "equation"? If so, how would I go about it? For example, if I am given the equation (x^2)/1-(y^2)/25=1, what process would I have to use to find the Parametric equations?

Thank You.

 

[hypermorphism]

If either variable was a function of the other, you could do the trivial way of letting one variable be the parameter. In any other case, there is no general way for generating a parametrization other than experience.
For example, knowing the identity cosh²(t) - sinh²(t) = 1 shows us that one parametrization of the curve is x = cosh(t), y=5*sinh(t). The similarity to a certain parametrization of an ellipse makes their names as "hyperbolic" neighbors to sine and cosine clear.

 

[tacman]

Yes, it is possible to transform an "x-y" equation into a parametric equation. The process involves expressing the variables x and y in terms of a third variable, typically denoted by t. This third variable, also known as the parameter, will be used to define the coordinates of points on the curve.

To find the parametric equations for the given equation (x^2)/1-(y^2)/25=1, we can follow these steps:

1. First, solve for x in terms of y by rearranging the equation to get: x = ±√(1 - (y^2)/25).

2. Next, we can define x and y in terms of t by substituting x = ±√(1 - (y^2)/25) into the equation x = x(t) and y = y(t).

3. By choosing a suitable range for t, we can now obtain the parametric equations by substituting the expressions for x and y into the equations x = x(t) and y = y(t). For example, if we let t range from 0 to 2π, we can obtain the parametric equations: x = 5cos(t) and y = 5sin(t).

In general, the process of finding parametric equations involves expressing the variables x and y in terms of a third variable t, and then defining x and y as functions of t. The range of t can be chosen to fit the desired domain of the curve. I hope this helps!