Consider the parametric equation `x=(a(1-t^(2)))/(1+t^(2))" and "y=(2at)/(1+t^(2))`
What does the equation represent ?

Options

A. It represents a circle of diameter a
B. It represents a circle of radius a
C. It represents a parabola
D. None of the above

Answer

To determine what the given parametric equations represent, we start with the equations:

1. \( x = \frac{a(1 - t^2)}{1 + t^2} \)
2. \( y = \frac{2at}{1 + t^2} \)

### Step 1: Substitute \( t \) with \( \tan(\theta) \)

We can express \( t \) in terms of \( \theta \) using the substitution \( t = \tan(\theta) \). This gives us:

- \( t^2 = \tan^2(\theta) \)

Now substituting \( t \) into the equations for \( x \) and \( y \):

\[ x = \frac{a(1 - \tan^2(\theta))}{1 + \tan^2(\theta)} \]

\[ y = \frac{2a\tan(\theta)}{1 + \tan^2(\theta)} \]

### Step 2: Simplify the expressions for \( x \) and \( y \)

Using the identity \( 1 - \tan^2(\theta) = \cos(2\theta) \) and \( 1 + \tan^2(\theta) = \sec^2(\theta) \):

\[ x = a \cdot \cos(2\theta) \]

\[ y = 2a \cdot \frac{\tan(\theta)}{1 + \tan^2(\theta)} = 2a \cdot \sin(2\theta) \]

### Step 3: Use trigonometric identities

Now we can rewrite \( y \):

\[ y = a \cdot \sin(2\theta) \]

### Step 4: Relate \( x \) and \( y \)

Now we have:

\[ x = a \cdot \cos(2\theta) \]
\[ y = a \cdot \sin(2\theta) \]

### Step 5: Find the relationship between \( x \) and \( y \)

Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):

\[ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{a}\right)^2 = \cos^2(2\theta) + \sin^2(2\theta) = 1 \]

### Step 6: Write the final equation

Thus, we can express this as:

\[ \frac{x^2}{a^2} + \frac{y^2}{a^2} = 1 \]

This is the equation of a circle with radius \( a \).

### Conclusion

The parametric equations represent a circle of radius \( a \).

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