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Express the following ellipses in general form and sketch its graphs.

You provided the parametric equations:

\[
x = 3\cos\theta + 3, \quad y = 5\sin\theta - 5
\]

### Step 1: Eliminate the parameter \(\theta\)

We aim to express this as an equation in terms of \(x\) and \(y\). Begin by isolating \(\cos\theta\) and \(\sin\theta\) in the parametric equations:

1. From \(x = 3\cos\theta + 3\):
   \[
   \cos\theta = \frac{x - 3}{3}
   \]

2. From \(y = 5\sin\theta - 5\):
   \[
   \sin\theta = \frac{y + 5}{5}
   \]

### Step 2: Use the identity \(\sin^2\theta + \cos^2\theta = 1\)

Substitute the expressions for \(\cos\theta\) and \(\sin\theta\) into the identity:

\[
\left(\frac{x - 3}{3}\right)^2 + \left(\frac{y + 5}{5}\right)^2 = 1
\]

### Step 3: Simplify

Expand the terms:

\[
\frac{(x - 3)^2}{9} + \frac{(y + 5)^2}{25} = 1
\]

This is the equation of an ellipse in standard form:

\[
\frac{(x - 3)^2}{9} + \frac{(y + 5)^2}{25} = 1
\]

### Step 4: Analyze the Ellipse

- The center of the ellipse is at \((3, -5)\).
- The semi-major axis is 5 (since \(25 = 5^2\)).
- The semi-minor axis is 3 (since \(9 = 3^2\)).

This is a vertically oriented ellipse because the semi-major axis is along the \(y\)-axis.

### Step 5: Sketch the Graph

To sketch the ellipse:

1. Plot the center at \((3, -5)\).
2. From the center, move 5 units up and down along the \(y\)-axis (since the semi-major axis is 5).
3. Move 3 units left and right along the \(x\)-axis (since the semi-minor axis is 3).
4. Draw the ellipse through these points.

Would you like me to generate a detailed graph for this ellipse, or provide further details on the derivation?

### Follow-up Questions:

1. Can you describe the key characteristics of an ellipse based on its equation?
2. How do we interpret the semi-major and semi-minor axes of an ellipse?
3. What is the importance of the center in the standard form of an ellipse?
4. How does the orientation of the axes affect the shape of the ellipse?
5. What is the effect of changing the coefficients in the equation of the ellipse?

### Tip:

Always remember that the semi-major axis is the longest diameter of the ellipse, and the semi-minor axis is the shortest.

Learn how to express parametric equations of an ellipse in general form and sketch the graph step by step. This example involves trigonometric identities and parametric equations to form an ellipse, ideal for high school students in Grades 10-12.

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