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options are a. a pairof straight lines b. an ellipse c. a parabola d. a hyperbola

The problem gives the parametric equations of a curve as:
\[
x = u^2 + u + 1 \quad \text{and} \quad y = u^2 - u + 1
\]
We are asked to determine the type of curve these equations represent.

### Step 1: Express \(x\) and \(y\) in terms of \(u\)
Given the equations:
\[
x = u^2 + u + 1
\]
\[
y = u^2 - u + 1
\]

### Step 2: Eliminate the parameter \(u\)
Subtract the second equation from the first:
\[
x - y = (u^2 + u + 1) - (u^2 - u + 1)
\]
\[
x - y = u + u = 2u
\]

Thus, we have:
\[
u = \frac{x - y}{2}
\]

### Step 3: Substitute \(u\) back into one of the original equations
Substitute \(u = \frac{x - y}{2}\) into the equation for \(x\):
\[
x = \left(\frac{x - y}{2}\right)^2 + \frac{x - y}{2} + 1
\]

Expand and simplify:
\[
x = \frac{(x - y)^2}{4} + \frac{x - y}{2} + 1
\]

Multiply the entire equation by 4 to eliminate the fraction:
\[
4x = (x - y)^2 + 2(x - y) + 4
\]

Now expand \((x - y)^2\):
\[
4x = x^2 - 2xy + y^2 + 2x - 2y + 4
\]

### Step 4: Rearrange the equation
To determine the type of curve, let's rearrange this into a standard conic form:
\[
x^2 - 2xy + y^2 + 2x - 2y + 4 - 4x = 0
\]
\[
x^2 - 2xy + y^2 - 2x - 2y + 4 = 0
\]

### Step 5: Identify the curve
The equation we obtained is quadratic and includes a cross term \(-2xy\), which is characteristic of a conic section. Specifically, because of the presence of both \(x^2\) and \(y^2\) terms with equal coefficients and a non-zero cross term \(-2xy\), this is the equation of a **hyperbola**.

### Final Answer:
The curve described by the parametric equations is a **hyperbola**, corresponding to option **d. a hyperbola**.

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Do you want more details or have any questions? Here are some related questions:

1. How can you distinguish between different conic sections using their general quadratic equation?
2. What is the significance of the discriminant in determining the type of conic section?
3. How would the parametric equations change if the curve were an ellipse?
4. Can the equation of a hyperbola always be reduced to its standard form by rotation?
5. How do cross terms like \(-2xy\) indicate a rotation of the coordinate axes?

**Tip:** When dealing with conic sections, recognizing the presence of cross terms like \(xy\) and their coefficients can quickly indicate the type of curve, especially when determining whether the curve is a hyperbola.

Learn how to identify the type of curve given parametric equations. This example demonstrates the process of determining a hyperbola from parametric equations using algebraic transformations and standard conic section forms.

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