Question

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

• A. hyperbola
• B. ellipse
• C. parabola
• D. rectangular hyperbola

Answer 1

The correct option is C parabola

The equations $x=t^2+t+1$ and $y=t^2-t+1$

Subtracting $y$ from $x,$ we get

$x-y=2t$

$\Rightarrow$ $t=\frac{x-y}{2}$ ----- ( 1 )

Now, adding $x$ and $y$ we get,

$\Rightarrow$ $x+y=2t^2+2$ ----- ( 2 )

Substituting the value of $t$ from equation ( 1 ) in equation ( 2 ), we get

$x+y=2{\left(\frac{x-y}{2}\right)}^2+2$

$\Rightarrow$ $x+y=2\times \frac{x^2+y^2-2xy}{4}+2$

$\Rightarrow$ $x+y=\frac{x^2+y^2-2xy}{2}+2$

$\Rightarrow$ $2x+2y=x^2+y^2-2xy+4$

$\Rightarrow$ $x^2+y^2-2xy+4-2x-2y=0$ ----- ( 3 )

The general second degree equation is of the form

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ ----- ( 4 )

Comparing equations ( 3 ) and ( 4 ), we have

$a=1,h=-1,b=1,g=-1,f=-1$ and $c=4$

This equation represents parabola if $h^2-ab=0$

$h^2-ab=(-1{)}^2-(1\left)\right(1)=1-1=0$

$\therefore$ The equation ( 3 ) represents a parabola.