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To convert the given polar equations into rectangular form, we will use the following relationships between polar and rectangular coordinates:

1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)
3. \( r^2 = x^2 + y^2 \)
4. \( \tan \theta = \frac{y}{x} \)

Now let's solve the two parts:

### Part (a)
The given polar equation is:

\[
r = \frac{2}{4 \cos \theta - 5 \sin \theta}
\]

To convert to rectangular form, we multiply both sides of the equation by \( (4 \cos \theta - 5 \sin \theta) \):

\[
r(4 \cos \theta - 5 \sin \theta) = 2
\]

Now substitute \( r \cos \theta = x \) and \( r \sin \theta = y \):

\[
4x - 5y = 2
\]

This is the rectangular form of the equation.

### Part (b)
The given polar equation is:

\[
r = -10 \sin \theta
\]

Using the substitution \( r \sin \theta = y \), the equation becomes:

\[
r = -10 \cdot \frac{y}{r}
\]

Now, multiply both sides by \( r \):

\[
r^2 = -10y
\]

Using the identity \( r^2 = x^2 + y^2 \):

\[
x^2 + y^2 = -10y
\]

Thus, the rectangular form is:

\[
x^2 + y^2 + 10y = 0
\]

We could further simplify or complete the square if necessary, but this is the basic rectangular form.

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Do you have any questions or would you like further details? 

Here are 5 related questions:
1. How do you complete the square to simplify part (b) further?
2. What are the advantages of converting polar equations to rectangular form?
3. Can you plot both the rectangular and polar forms to see the curve behavior?
4. How do you derive the polar to rectangular coordinate transformations?
5. Can polar equations always be converted to rectangular form?

**Tip:** It's helpful to remember that \( r = \sqrt{x^2 + y^2} \) and \( \tan \theta = \frac{y}{x} \) when converting between forms!

Convert each polar equation to rectangular form:
(a) r = 2 / (4 cos θ - 5 sin θ)
(b) r = -10 sin θ

Learn how to convert polar equations to rectangular form with step-by-step instructions. We take polar equations like r = 2/(4 cos θ - 5 sin θ) and r = -10 sin θ, and apply the transformations to derive their rectangular form. Suitable for Grades 10-12.

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