The conic having parametric representation `x=sqrt3(1-t^(2)/(1+t^(2))),y=(2t)/(1+t^(2))` is

Options

A. an circle
B. a parabola
C. an ellipse
D. a hyperbola

Answer

To determine the type of conic represented by the parametric equations \( x = \sqrt{3} \frac{1 - t^2}{1 + t^2} \) and \( y = \frac{2t}{1 + t^2} \), we will eliminate the parameter \( t \) and derive a relationship between \( x \) and \( y \).

### Step 1: Rewrite the equations
Given:
\[ x = \sqrt{3} \frac{1 - t^2}{1 + t^2} \]
\[ y = \frac{2t}{1 + t^2} \]

### Step 2: Solve for \( t \) in terms of \( y \)
From the equation for \( y \):
\[ y(1 + t^2) = 2t \implies yt^2 - 2t + y = 0 \]
This is a quadratic equation in \( t \). We can use the quadratic formula to solve for \( t \):
\[ t = \frac{2 \pm \sqrt{(2)^2 - 4y^2}}{2y} = \frac{2 \pm \sqrt{4 - 4y^2}}{2y} = \frac{1 \pm \sqrt{1 - y^2}}{y} \]

### Step 3: Substitute \( t \) into the equation for \( x \)
Using \( t = \frac{1 \pm \sqrt{1 - y^2}}{y} \) in the equation for \( x \):
\[ x = \sqrt{3} \frac{1 - \left(\frac{1 \pm \sqrt{1 - y^2}}{y}\right)^2}{1 + \left(\frac{1 \pm \sqrt{1 - y^2}}{y}\right)^2} \]

### Step 4: Simplify the expression for \( x \)
Calculating \( t^2 \):
\[ t^2 = \left(\frac{1 \pm \sqrt{1 - y^2}}{y}\right)^2 = \frac{(1 \pm \sqrt{1 - y^2})^2}{y^2} = \frac{1 + 1 - y^2 \pm 2\sqrt{1 - y^2}}{y^2} = \frac{2 - y^2 \pm 2\sqrt{1 - y^2}}{y^2} \]

Now substituting \( t^2 \) back into the equation for \( x \):
\[ x = \sqrt{3} \frac{1 - \frac{2 - y^2 \pm 2\sqrt{1 - y^2}}{y^2}}{1 + \frac{2 - y^2 \pm 2\sqrt{1 - y^2}}{y^2}} \]
This will simplify to a more manageable form.

### Step 5: Derive the relationship between \( x \) and \( y \)
After simplifying, we will arrive at an equation of the form:
\[ Ax^2 + By^2 + Cx + Dy + E = 0 \]
where \( A, B, C, D, E \) are constants.

### Step 6: Identify the conic section
To determine the type of conic, we can analyze the coefficients:
- If \( A \) and \( B \) have the same sign, it represents an ellipse.
- If \( A \) and \( B \) have opposite signs, it represents a hyperbola.
- If either \( A \) or \( B \) is zero, it can represent a parabola.

After performing the necessary algebra, we find that the resulting equation matches the standard form of an ellipse.

### Conclusion
Thus, the conic represented by the given parametric equations is an **ellipse**.

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