Put into polar coordinates, y^2=4ax, with the origin as a pole.

The equation \( y^2 = 4ax \) is the equation of a parabola in Cartesian coordinates. To convert this into polar coordinates, we use the relationships:

\[
x = r \cos \theta \quad \text{and} \quad y = r \sin \theta
\]

Substitute these into the given equation:

\[
(r \sin \theta)^2 = 4a(r \cos \theta)
\]

This simplifies to:

\[
r^2 \sin^2 \theta = 4ar \cos \theta
\]

Now, divide both sides by \( r \) (assuming \( r \neq 0 \)):

\[
r \sin^2 \theta = 4a \cos \theta
\]

Therefore, the equation in polar coordinates is:

\[
r = \frac{4a \cos \theta}{\sin^2 \theta}
\]

This is the polar form of the parabola \( y^2 = 4ax \) with the origin as the pole.

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do you convert a general conic section to polar coordinates?
2. What is the geometric interpretation of this parabola in polar coordinates?
3. How would you find the directrix of the parabola \( y^2 = 4ax \) in polar form?
4. What happens to the equation if you move the origin to a different point?
5. How can this transformation help in graphing polar curves?

**Tip:** Always remember to substitute \( x \) and \( y \) with \( r \cos \theta \) and \( r \sin \theta \) respectively when converting between Cartesian and polar coordinates.

Put into polar coordinates, y^2 = 4ax, with the origin as a pole.

This detailed solution explains how to convert the Cartesian equation y^2 = 4ax into polar coordinates using x = r cos θ and y = r sin θ. Learn key concepts of polar transformations and parabolic equations. Ideal for students in Grades 10-12.

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