Problem 85 - Calculus Early Transcendentals

Question

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=t, y=\sqrt{4-t^{2}}$$

Step-by-Step Solution

Answer

Question: Eliminate the parameter \(t\) from the given parametric equations \(\begin{cases} x = t \\ y = \sqrt{4-t^2} \end{cases}\) and express in the form of a single equation in \(x\) and \(y\). Answer: \(y=\sqrt{4-x^2}\)

Step 1: Express t in terms of x

From the given parametric equation, we have \(x = t.\) Therefore, we can write \(t = x.\)

Step 2: Substitute the expression of t in the second equation

Now, substitute the expression of \(t\) from step 1 into the second equation: $$y=\sqrt{4-t^{2}}$$ Substitute \(t = x\): $$y=\sqrt{4-x^2}$$

Step 3: Verify the resulting equation

The resulting equation is: $$y=\sqrt{4-x^2}$$ This equation is in the form of a circle. We can now see that we have eliminated the parameter \(t\) and expressed the parametric equations as a single equation in \(x\) and \(y\).

Key Concepts

Eliminate the Parameter

Parametric equations often involve parameters like \(t\) that define the relationship between \(x\) and \(y\) in a pair of equations. Eliminating the parameter is the process of rewriting these equations without the parameter. Essentially, it means we want to express the relationship between \(x\) and \(y\) without involving \(t\).

To eliminate the parameter, we identify one of the equations to express \(t\) in terms of \(x\). For the given parametric equations, we have:

• \(x = t\)
• \(y = \sqrt{4 - t^2}\)

From the equation \(x = t\), it's straightforward to solve for \(t\) by rearranging it as \(t = x\), which is simple and direct. This substitution allows us to eliminate \(t\) and rewrite the second equation directly in terms of \(x\) and \(y\) only.

Equation in x and y

With \(t\) eliminated, our aim is to express a direct relationship between \(x\) and \(y\). So, using the expression \(t = x\) found from the first parametric equation, we substitute this into the second equation. Originally, the second equation is given by:

• \(y = \sqrt{4 - t^2}\)

By substituting \(t = x\), this becomes:

• \(y = \sqrt{4 - x^2}\)

This is the equation in terms of \(x\) and \(y\), representing the relationship originally described by the parametric equations. Thus, we have successfully expressed how \(x\) and \(y\) are related without involving the parameter \(t\), simplifying the given set of equations into a single concise equation.

Circle Equation

The resulting equation \(y = \sqrt{4-x^2}\) resembles a part of the equation of a circle. In coordinate geometry, a circle with center at the origin \((0,0)\) and a radius \(r\) is represented as:

• \(x^2 + y^2 = r^2\)

In this case, if we rearrange the equation \(y^2 = 4 - x^2\), we get \(x^2 + y^2 = 4\). This equation fits the standard circle equation with \(r = 2\), which represents a circle of radius 2 centered at the origin.

However, it’s important to note that \(y = \sqrt{4 - x^2}\) specifically describes the upper half of this circle. This is because the square root function returns only the positive values, which restricts \(y\) to being non-negative. Consequently, this equation captures the semi-circle lying above the x-axis, where \(y \geq 0\).