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You provided two polar coordinate equations, and the task is to convert them into Cartesian coordinates.

### Given Equations:
- a. \( r \cos \theta = 6 \)
- b. \( r^2 - 8r \cos \theta - 4r \sin \theta + 11 = 0 \)

### Conversion Relations Between Polar and Cartesian Coordinates:
1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)
3. \( r^2 = x^2 + y^2 \)

### Solution:

#### a. \( r \cos \theta = 6 \)
Using \( x = r \cos \theta \), we can directly substitute:

\[
x = 6
\]

Thus, in Cartesian coordinates, the equation becomes:
\[
x = 6
\]
This represents a vertical line passing through \( x = 6 \).

#### b. \( r^2 - 8r \cos \theta - 4r \sin \theta + 11 = 0 \)
Now, apply the substitutions:
- \( r^2 = x^2 + y^2 \)
- \( r \cos \theta = x \)
- \( r \sin \theta = y \)

Substituting into the equation:

\[
x^2 + y^2 - 8x - 4y + 11 = 0
\]

This is a quadratic equation in terms of \( x \) and \( y \), representing a circle or another conic section. To simplify, we can rearrange the terms:

\[
x^2 - 8x + y^2 - 4y = -11
\]

Now, complete the square for both \( x \) and \( y \):
- For \( x \), \( x^2 - 8x \) becomes \( (x - 4)^2 - 16 \)
- For \( y \), \( y^2 - 4y \) becomes \( (y - 2)^2 - 4 \)

So the equation becomes:

\[
(x - 4)^2 - 16 + (y - 2)^2 - 4 = -11
\]
Simplifying further:

\[
(x - 4)^2 + (y - 2)^2 = 9
\]

This is the equation of a circle with center \( (4, 2) \) and radius 3.

### Final Cartesian Forms:
- a. \( x = 6 \)
- b. \( (x - 4)^2 + (y - 2)^2 = 9 \)

Would you like more details or have any questions about these solutions?

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Here are 5 related questions:
1. How do you complete the square for a general quadratic equation?
2. What are the key properties of a circle in Cartesian coordinates?
3. How can you identify a conic section from a quadratic equation in \(x\) and \(y\)?
4. How do polar coordinates relate to Cartesian coordinates geometrically?
5. What is the general form of a polar-to-Cartesian coordinate transformation?

**Tip:** When converting polar equations to Cartesian form, always use the key identities \(x = r \cos \theta\) and \(y = r \sin \theta\).

Ubah ke sistem koordinat kartesius persamaan: a. r cos(θ) = 6, b. r^2 - 8r cos(θ) - 4r sin(θ) + 11 = 0

Learn how to convert polar equations to Cartesian coordinates with detailed step-by-step solutions. Problems include r cos(θ) = 6 and r^2 - 8r cos(θ) - 4r sin(θ) + 11 = 0. Understand how to transform polar equations into x and y forms using core algebraic and trigonometric concepts. Suitable for high school students.

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