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Grade 11Analytical Geometry

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

Swagat Panda

10 Years agoGrade 11

2 Answers

Vijay Mukati

10 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

10 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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