Put into polar coordinates y=4ax, with the origin as a pole.

To convert the equation \( y = 4ax \) into polar coordinates, we use the following transformations:

- \( x = r \cos \theta \)
- \( y = r \sin \theta \)

Here, \( r \) is the distance from the origin (the pole), and \( \theta \) is the angle with the positive \( x \)-axis.

### Step-by-step conversion:
1. Start with the given equation:
   \[
   y = 4ax
   \]
2. Substitute the polar coordinate expressions for \( x \) and \( y \):
   \[
   r \sin \theta = 4a \cdot (r \cos \theta)
   \]
3. Simplify the equation:
   \[
   r \sin \theta = 4ar \cos \theta
   \]
4. Divide both sides by \( r \) (assuming \( r \neq 0 \)):
   \[
   \sin \theta = 4a \cos \theta
   \]
5. Solve for \( r \) to obtain the polar form. Rearrange this equation:
   \[
   \tan \theta = 4a
   \]
   This expresses the relationship between the angle \( \theta \) and the constant \( a \), but it doesn't explicitly give \( r \) as a function of \( \theta \). Therefore, in polar form, the equation simplifies to:
   \[
   \theta = \tan^{-1}(4a)
   \]

Thus, the equation \( y = 4ax \) in polar coordinates is described by the angle \( \theta \), where \( \theta = \tan^{-1}(4a) \).

Would you like further details or examples?

### 5 Related Questions:
1. How do you convert a general linear equation into polar coordinates?
2. What are the polar forms of common curves like circles or parabolas?
3. Can we express this equation parametrically in polar coordinates?
4. How do you find the slope of a line in polar coordinates?
5. What is the general process to convert conic sections into polar form?

**Tip:** In polar coordinates, always check if you need to express \( r \) or \( \theta \) explicitly depending on the equation's form.

Put into polar coordinates y = 4ax, with the origin as a pole.

Learn how to convert the equation y = 4ax into polar coordinates, using key transformations such as x = r cos(θ) and y = r sin(θ). This solution includes step-by-step instructions, covering key concepts in polar coordinate transformation. Suitable for students in Grades 11-12.

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