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Q)

The curve disturbed parametrically by $x=t^2+t+1\;y=t^2-t+1$ represents

$\begin{array}{1 1}(A)\;\text{a pair of straight line} \\(B)\;\text{an ellipse} \\ (C)\;\text{a parabola} \\(D)\;\text{a hyperbola} \end{array}$

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A)
$x=t^2+t+1$
$y=t^2-t+1$
Eliminate the parameter t from (1) as follows
$x-y=2t$
$x= \bigg(\large\frac{x-y}{2}\bigg)^2+\bigg(\large\frac{x-y}{2} \bigg)$$+1$
=> $4x=(x-y)^2+2x-2y+4$
=> $ (x-y)^2=2(x+y-2)$
Hence it is parabola
Hence C is the correct answer.
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