Question

If t be a variable quantity, find the locus of the point (x, y) when
$x=1+t+t^2$ and $y=1-t+t^2.$

Answer 1

Given: $x=1+t+t^2$ and $y=1-t+t^2$

To get the required locus, we are supposed to eliminate 't' from these equations.

$\Rightarrow$ $x=1+t+t^2$........eqn(i)

$\Rightarrow$ $y=1-t+t^2$........eqn(ii)

Adding the eqns (i) & (ii), we get

$\Rightarrow$ $x+y=2(1+t^2)$........eqn(iii)

Subtracting eqn (i) & (ii) gives,

$\Rightarrow$ $x-y=2t$..........eqn(iv)

Substitute 't' from eq(iv) in eqn(iii)

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

$\Rightarrow$ $(x-y{)}^2-2(x+y)+4=0$

This is the required locus of the point (x,y).