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(C) Given,$x = t^2 + t + 1 \quad (i)$ and $y = t^2 - t + 1 \quad (ii)$.Adding $(i)$ and $(ii)$,we get $x + y = 2t^2 + 2 = 2(t^2 + 1) \quad (iii)$.Subt

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$A$ parabola

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(C) Given,$x = t^2 + t + 1 \quad (i)$ and $y = t^2 - t + 1 \quad (ii)$.Adding $(i)$ and $(ii)$,we get $x + y = 2t^2 + 2 = 2(t^2 + 1) \quad (iii)$.Subtracting $(ii)$ from $(i)$,we get $x - y = 2t$,which implies $t = \frac{x - y}{2} \quad (iv)$.Substituting $(iv)$ into $(iii)$:$x + y = 2 \left[ \left( \frac{x - y}{2} \right)^2 + 1 \right]$$x + y = 2 \left[ \frac{x^2 + y^2 - 2xy}{4} + 1 \right]$$x + y = \frac{x^2 + y^2 - 2xy}{2} + 2$$2x + 2y = x^2 + y^2 - 2xy + 4$$x^2 - 2xy + y^2 - 2x - 2y + 4 = 0$This is of the form $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$,where $a=1, h=-1, b=1, g=-1, f=-1, c=4$.The discriminant $h^2 - ab = (-1)^2 - (1)(1) = 1 - 1 = 0$.Since $h^2 - ab = 0$,the equation represents a parabola.

What does the curve defined parametrically by $x = t^2 + t + 1$ and $y = t^2 - t + 1$ represent?

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