The curve described parametrically by `x=t^2+t+1,y=t^2-t+1` represents

Options

A. a pair of straight lines
B. an ellipse
C. a parabola
D. a hyperbola

Answer

To determine the type of curve described by the parametric equations \( x = t^2 + t + 1 \) and \( y = t^2 - t + 1 \), we will follow these steps:

### Step 1: Express \( t \) in terms of \( x \) and \( y \)
We start with the equations:
1. \( x = t^2 + t + 1 \) (Equation 1)
2. \( y = t^2 - t + 1 \) (Equation 2)

### Step 2: Subtract Equation 2 from Equation 1
To eliminate \( t^2 \), we subtract Equation 2 from Equation 1:
\[ x - y = (t^2 + t + 1) - (t^2 - t + 1) \]
This simplifies to:
\[ x - y = 2t \]
From this, we can express \( t \):
\[ t = \frac{x - y}{2} \]

### Step 3: Substitute \( t \) back into one of the original equations
Now, we substitute \( t \) back into Equation 1 to express \( x \) in terms of \( y \):
\[ x = \left(\frac{x - y}{2}\right)^2 + \left(\frac{x - y}{2}\right) + 1 \]

### Step 4: Simplify the equation
Expanding the equation:
\[ x = \frac{(x - y)^2}{4} + \frac{x - y}{2} + 1 \]
Multiplying through by 4 to eliminate the fraction:
\[ 4x = (x - y)^2 + 2(x - y) + 4 \]
This simplifies to:
\[ 4x = (x - y)^2 + 2x - 2y + 4 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ (x - y)^2 - 2y + 4 - 4x = 0 \]
This can be rewritten as:
\[ (x - y)^2 - 4x - 2y + 4 = 0 \]

### Step 6: Recognizing the form of the equation
This equation can be rearranged further to show that it is indeed a conic section. The general form of a conic section is:
\[ Ax^2 + By^2 + 2Hxy + 2Gx + 2Fy + C = 0 \]
By comparing coefficients, we can determine the type of conic.

### Step 7: Determine the type of conic
To determine the type of curve, we compute the discriminant \( \Delta \):
\[ \Delta = AB - H^2 \]
In our case, we find that \( A = 1 \), \( B = 1 \), and \( H = -1 \). Therefore:
\[ \Delta = 1 \cdot 1 - (-1)^2 = 1 - 1 = 0 \]
Since \( \Delta = 0 \), this indicates that the curve is a parabola.

### Conclusion
The curve described by the parametric equations \( x = t^2 + t + 1 \) and \( y = t^2 - t + 1 \) represents a parabola.

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