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

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

Grade:11

2 Answers

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.

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