Question

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

We have, $x=t^2+t+1$ .... (i)
and $y=t^2-t+1$ .... (ii)
Now, $x+y=2(1+t^2)$ .... (iii)
and $x-y=2t$ .... (iv)
Now, from Eqs. (iii) and (iv), we get
$x+y=2[1+{\left(\frac{x-y}{2}\right)}^2]$
$\Rightarrow x+y=2\left[\frac{4+x^2+y^2-2xy}{4}\right]$
$\Rightarrow x^2+y^2-2xy-2x-2y+4=0$ .... (v)
On comparing with, we get
$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$
We get, $a=1,b=1,c=4,h=-1,g=-1,f=-1$
$△=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$
Now, $△=1\cdot 1\cdot 4+2(-1)(-1)(-1)-1\times (-1{)}^2-1\times (-1{)}^2-4(-1{)}^2$
$=4-2-1-1-4$
$=-4$, therefore, $△\ne 0$
and $ab-h^2=1\cdot 1-(1{)}^2=1-1=0$
So, it is equation of a parabola.