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?

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 $y = r \sin \theta$.