https://file.aiquickdraw.com/mathbot/file/1729043200893rbm5ifbj.jpg

Let's solve the problem using the given parametric equations:

\[
x = \ln(5t^2 + 1), \quad y = \frac{t}{t+5}
\]

We need to find the points \((x, y)\) for \(t = 0\).

1. **For \(x\):**

   \[
   x = \ln(5(0)^2 + 1) = \ln(1) = 0
   \]

2. **For \(y\):**

   \[
   y = \frac{0}{0 + 5} = 0
   \]

Therefore, the point corresponding to \(t = 0\) is \((x, y) = (0, 0)\).

Would you like details on other values of \(t\) or have any other questions?

Here are some related questions to consider:
1. What are the points for \(t = -2, -1, 1, 2\)?
2. How do logarithmic functions behave as their argument approaches zero?
3. What happens to the parametric equation \(y = \frac{t}{t+5}\) as \(t\) approaches large positive or negative values?
4. How can we plot the parametric curve for this range of \(t\)?
5. What would happen if we changed the parameters in the equations?

**Tip:** Remember, the natural logarithm function, \(\ln(x)\), is only defined for positive values of \(x\).

For the given parametric equations, find the points (x, y) corresponding to the parameter values t = -2, -1, 0, 1, 2. The parametric equations are: x = ln(5t^2 + 1), y = t / (t + 5).

Solve the parametric equations x = ln(5t^2 + 1) and y = t / (t + 5) for values of t = -2, -1, 0, 1, 2. This problem explores the use of logarithmic and rational functions in parametric form, suitable for advanced high school students.

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