Question

The conic section represented by the equations $x=t^2+t+1$ and $y=t^2-t+1$ is ________ .

• A. hyperbola
• B. point
• C. parabola
• D. circle

Answer 1

The correct option is C parabola

Given equations are $x=t^2+t+1$ and $y=t^2-t+1$.
$x+y=2t^2+2$ and $x-y=2t$
$x+y=2t^2+2\Rightarrow 2\left(x+y\right)={\left(2t\right)}^2+4\Rightarrow 2\left(x+y\right)={\left(x-y\right)}^2+4\Rightarrow x^2+y^2-2\mathrm{xy}-2x-2y+4=0$
Here, a = 1, b = 1, h = -1, g = -1, f = -1, c = 4.
Now, $△=\mathrm{abc}+2\mathrm{fgh}-{\mathrm{af}}^2-{\mathrm{bg}}^2-{\mathrm{ch}}^2\Rightarrow △=-4\ne 0$And $h^2=\mathrm{ab}$
So, the given equation represents a parabola.