Question

The curve described parametrically by, $x=t^2+t+1,y=t^2-t+1$ represents -

• A. A pair of straight lines
• B. An ellipse
• C. A parabola
• D. A hyperbola

Answer 1

The correct option is C A parabola

$x=t^2+t+1$

$y=t^2-t+1$

Therefore,

$x-y=2t.....\left(1\right)$

$x+y=2t^2+2$

$\Rightarrow 2(x+y)={\left(2t\right)}^2+4$

$\Rightarrow 2(x+y)={(x-y)}^2+4\left[\text{From}\left(1\right)\right]$

$\Rightarrow {(x-y)}^2=2(x+y)-4$

$\Rightarrow Y^2=2X-4$

Where as,

$Y=x-y$

$X=x+y$

Hence the curve describes a parabola.