Question

The locus of the point $x=t+\frac{1}{t}$ and $y=t-\frac{1}{t}$ is

• A. a straight line
• B. a circle
• C. a parabola
• D. a hyperbola

Answer 1

The correct option is D a hyperbola

Given parametric equations are
$x=t+\frac{1}{t}\cdots \left(1\right)y=t-\frac{1}{t}\cdots \left(2\right)$
From $(1{)}^2-(2{)}^2,$ we get
$x^2-y^2=4$
On comparing with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0,$
we have
$a=1,h=0,b=-1,g=f=0,c=-4$
$△=abc+2fgh-a{f}^2-b{g}^2-c{h}^2=4\ne 0$
and, $h^2>ab$
$\Rightarrow$ The locus is a hyperbola.