Question

If the parabola $x^2=2(2x+y)$ is shifted $2$ unit along negative x-axis and $2$ units along positive y-axis, then what will be the vertex of new parabola?

Answer 1

Given the equation of parabola $x^2=2(2x+y)$
On simplifying we get,
$\Rightarrow x^2=4x+2y$
$\Rightarrow x^2-4x=2y$
On adding $4$ both side we get,
$\Rightarrow x^2-4x+4=2y+4$
$\Rightarrow (x-2{)}^2=2(y+2)$
This is the standard equation of parabola we got on simplifying.

Now, translating it $2$ units along negative x-axis and $2$ unit along positive y-axis we get,
$\Rightarrow (x-2+2{)}^2=2(y+2-2)$
$\Rightarrow x^2=2y$
On comparing $x^2=2y$ with translated equation of parabola $(x-h{)}^2=4a(y-k)$ we get the coordinate of vertex as $(0,0).$