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# The curve described parametrically by x = t2 + t + 1, y = t2– t + 1 represents?

Vijay Mukati
askIITians Faculty 2590 Points
5 years ago
Dear Student,
Solve this equations to eliminate the t. You will get a second degree equation in x and y. Then compare with the general equation of second degree equation i.e. ax2+by2+2hxy+2gx+2fy+c=0. Then you will find that
Since, abc+2fgh-af2-bg2-ch2 is not equal to zero and h2=ab. Therefore this equation will represent a parabola.
Thanks.
jagdish singh singh
173 Points
5 years ago
$\hspace{-0.6 cm}Given \bf{x=t^2+t+1\;\;,y=t^2-t+1\;,} Now \bf{x+y=2(t^2+1)}\\\\ and \bf{x-y=2t\;\;.} So \bf{2(x+y)=(2t)^2+4=(x-y)^2+4.}\\\\So we get \bf{(x-y)^2=2(x+y)+4\;,} Now \bf{x+y=X'\;\;,x-y=Y'}\\\\So we get \bf{Y'^2=2X'+4=2\left[X'+0.5\right]\Rightarrow Y'^2=2\left(X'+\frac{1}{2}\right)}. \\\\ So It represent Parabola.$