31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=t,y= √(4−t²) a

Verified step by step guidance

1. Start with the given parametric equations: \(x = t\) and \(y = \sqrt{4 - t^2}\).
2. Since \(x = t\), you can express \(t\) in terms of \(x\) as \(t = x\).
3. Substitute \(t = x\) into the equation for \(y\): \(y = \sqrt{4 - x^2}\).
4. To eliminate the square root, square both sides of the equation: \(y^2 = 4 - x^2\).
5. Rearrange the equation to express it in terms of \(x\) and \(y\): \(x^2 + y^2 = 4\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to remove the parameter t, resulting in a single equation relating x and y directly. This often requires solving one equation for t and substituting into the other.

Using Algebraic Manipulation and Identities

To eliminate the parameter, algebraic techniques such as isolating variables, squaring both sides, or applying identities (like Pythagorean identities) are used. These steps help transform parametric forms into standard Cartesian equations.