Question

Eliminate the parameter to express the following parametric equations as a single equation in $x$ and $y.$ $$x=\sqrt{t+1},y=\frac{1}{t+1}$$

Short answer

Question: Given the parametric equations $$x=\sqrt{t+1}$$ and $$y=\frac{1}{t+1}$$, eliminate the parameter `t` to express the equations as a single equation in terms of `x` and `y`. Answer: The single equation in terms of `x` and `y` is $$y=\frac{1}{x^2}.$$

Step-by-step solution

Solve one equation for t

We can start by solving the equation $$x=\sqrt{t+1}$$ for `t`. First, we will square both sides of the equation to eliminate the square root and then isolate `t` on one side.

Replace t in the second equation

Now, we will replace the value of `t` that we found in step 1 into the second equation, $$y=\frac{1}{t+1}$$, and simplify further to eliminate the parameter `t` and obtain a single equation in terms of `x` and `y`.

Simplify the equation in x and y

In this step, we will simplify the equation obtained in step 2 by multiplying both sides by the denominator to eliminate the fraction, and arrive at our final equation in terms of `x` and `y`.

Key concepts

Eliminate the Parameter

When dealing with parametric equations, a common task is to express the equations as a single equation in terms of purely the variables you care about, often $x$ and $y$. This process is known as eliminating the parameter. In the given problem, we have a parameter $t$ in the equations $x=\sqrt{t+1}$ and $y=\frac{1}{t+1}$. To eliminate $t$, we first need to express $t$ in terms of $x$ and then substitute this expression into the equation for $y$.

Steps to Eliminate the Parameter:

• Solve one of the parametric equations for the parameter $t$.
• Substitute $t$ into the other equation.
• Simplify to get a direct relationship between $x$ and $y$.

This way, we're left with a clear equation involving only the variables of interest, $x$ and $y$, free from the parameter.

Single Equation in x and y

After eliminating the parameter, we need to express the two equations as one single equation with only $x$ and $y$. This is the ultimate goal when transforming parametric equations, allowing us to see the relationship between these variables more directly. In our example, once $t$ is eliminated, we should be able to write an equation that directly relates $x$ with $y$.

This transformation is helpful because:

• It reduces the number of variables, making interpretation more straightforward.
• It highlights the implicit relationship between $x$ and $y$.
• It aids in visualizing the graph of the equation more easily as a typical Cartesian graph.

Knowing how to convert parametric equations into this form is a key skill in the study of mathematics and calculus.

Solve for t

Solving for the parameter $t$ is often the first and most crucial step in eliminating it and finding a relationship between $x$ and $y$. In this exercise, you start by isolating $t$ in one of the parametric equations.

For example, given $x=\sqrt{t+1}$, to find $t$:

• Square both sides to eliminate the square root: $x^2=t+1$.
• Isolate $t$ by subtracting 1 from both sides: $t=x^2-1$.

Now that you have $t$ expressed as $x^2-1$, substitute this expression back into the other equation, $y=\frac{1}{t+1}$. This allows you to eliminate $t$, producing a single equation in terms of $x$ and $y$, thereby completing the process.