Question

The equation \(r \cos \theta = 2a \sin^2 \theta\) represents the curve:

• A. \(x^3 = y^2(2a + x)\)
• B. \(x^2 = y^2(2a + x)\)
• C. \(x^3 = y^2(2a - x)\)
• D. \(x^3 = y^2(2a + x)\)

Answer 1

The Correct Option is C

The polar equation provided is:

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

Step 1: Conversion to Cartesian coordinates.

Recall the following polar-to-Cartesian transformations:

\[ x = r \cos \theta, \quad y = r \sin \theta, \quad r^2 = x^2 + y^2. \]

Substitute \(x = r \cos \theta\) into the initial equation:

\[ x = 2a \sin^2 \theta. \]

Using \(\sin^2 \theta = \frac{y^2}{r^2}\), substitute:

\[ x = 2a \cdot \frac{y^2}{r^2}. \]

Substitute \(r^2 = x^2 + y^2\):

\[ x = 2a \cdot \frac{y^2}{x^2 + y^2}. \]

Step 2: Equation simplification.

Multiply by \(x^2 + y^2\):

\[ x(x^2 + y^2) = 2ay^2. \]

Expand:

\[ x^3 + xy^2 = 2ay^2. \]

Rearrange:

\[ x^3 = y^2(2a - x). \]

Final Result: The equation of the curve is:

\[ \boxed{x^3 = y^2(2a - x)}. \]