Question

The curve represented by the parameters $x=t^2+1$ and $y=2t+1$ is

• A. a parabola
• B. a circle
• C. a hyperbola
• D. an ellipse

Answer 1

The correct option is A a parabola

Given parameters are
$x=t^2+1\Rightarrow t^2=x-1\cdots \left(1\right)y=2t+1\Rightarrow t=\frac{y-1}{2}\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$
$x-1=\frac{1}{4}(y-1{)}^2\Rightarrow y^2-2y-4x+5=0$
Now comparing it with general form of conic, we get
$a=0,b=1,h=0,c=5,g=-2,f=-1$
Now, $\Delta =0+0-0-4-0=-4\ne 0$
and $h^2=0=ab$
Hence, given conic represents a parabola.