The curve represented by the equations `x=sin^(2) theta, y=2 costheta` is

Options

A. ellipse
B. parabola
C. hyperbola
D. none

Answer

To determine the type of curve represented by the equations \( x = \sin^2 \theta \) and \( y = 2 \cos \theta \), we can follow these steps:

### Step 1: Express \(\sin^2 \theta\) and \(\cos^2 \theta\) in terms of \(x\) and \(y\)
From the given equations, we have:
- \( x = \sin^2 \theta \)
- \( y = 2 \cos \theta \)

### Step 2: Find \(\cos \theta\) in terms of \(y\)
From the equation \( y = 2 \cos \theta \), we can express \(\cos \theta\) as:
\[ \cos \theta = \frac{y}{2} \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the expressions we have:
\[ \sin^2 \theta + \left(\frac{y}{2}\right)^2 = 1 \]

### Step 4: Substitute \(x\) for \(\sin^2 \theta\)
Since \( x = \sin^2 \theta \), we can substitute \(x\) into the equation:
\[ x + \left(\frac{y}{2}\right)^2 = 1 \]

### Step 5: Simplify the equation
Now, let's simplify the equation:
\[ x + \frac{y^2}{4} = 1 \]
Multiplying through by 4 to eliminate the fraction:
\[ 4x + y^2 = 4 \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ y^2 = 4 - 4x \]
or
\[ y^2 = -4x + 4 \]

### Step 7: Identify the type of curve
The equation \(y^2 = -4x + 4\) can be rewritten as:
\[ y^2 = -4(x - 1) \]
This is in the standard form of a parabola \(y^2 = 4p(x - h)\) where \(h = 1\) and \(p = -1\). Therefore, this represents a parabola that opens to the left.

### Conclusion
Thus, the curve represented by the equations \(x = \sin^2 \theta\) and \(y = 2 \cos \theta\) is a parabola.

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