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The problem displayed involves parametric equations for a circle and asks you to eliminate the parameter $t$ to write the equation in standard form.

Let me transcribe the problem and the provided choices for better clarity. Please give me a moment.From the photo, the problem asks:

"Eliminate the parameter. Write the resulting equation in standard form."

The given parametric equations are:
$x = 2 + 2 \cos t, \quad y = 4 + 2 \sin t.$

The answer choices are:

1. $\frac{(x - 2)^2}{4} + \frac{(y - 4)^2}{4} = 1$
2. $(x - 2)^2 + \frac{(y - 4)^2}{4} = 1$
3. $\frac{(x - 2)^2}{4} + \frac{(y - 4)^2}{1} = 1$
4. $(x - 2)^2 + (y - 4)^2 = 1$.

Solution:

1. Start with the given equations:
   $x = 2 + 2 \cos t$ and $y = 4 + 2 \sin t$.

2. Rewrite them to isolate $\cos t$ and $\sin t$:
   $\cos t = \frac{x - 2}{2}, \quad \sin t = \frac{y - 4}{2}.$

3. Use the Pythagorean identity:
   $\cos^2 t + \sin^2 t = 1.$

4. Substitute $\cos t$ and $\sin t$ into the identity:
   $\left(\frac{x - 2}{2}\right)^2 + \left(\frac{y - 4}{2}\right)^2 = 1.$

5. Simplify:
   $\frac{(x - 2)^2}{4} + \frac{(y - 4)^2}{4} = 1.$

Final Answer:

The correct choice is Option 1: $\frac{(x - 2)^2}{4} + \frac{(y - 4)^2}{4} = 1$.

Would you like further clarification or assistance? Here are some related questions you might explore:

Related Questions:

1. What is the general form of the equation of a circle in parametric form?
2. How does the Pythagorean identity relate to parametric equations for a circle?
3. Can this approach be applied to parametric equations for ellipses or hyperbolas?
4. What are the key differences between standard and parametric forms of conic sections?
5. How do you derive the parametric equations for other shapes, such as ellipses?

Tip:

When converting parametric equations to standard form, always check for trigonometric identities (e.g., $\cos^2 t + \sin^2 t = 1$) to simplify the process.