Question

On the coordinates plane XOY hatch a set of points whose coordinates satisfy the following inequalities in two variables x and y (show the boundary belonging to the set by a solid line and that not belonging to the set by a dash line).
$|y|+2|x|⩽x^2+1$

Answer 1

R.E.F image

$|y|+2|x|\le x^2+1$

$\Rightarrow |y|\le x^2-2|x|+1$

$\Rightarrow$ we can write $|x{|}^2=x^2$

$|y|\le (|x|-1{)}^2$

Let's divide this into quadrants

$I^{st}$ qnad.

$x>0$ & $y>0$

$y\le (x-1{)}^2$

$2^{nd}$ qnad

$x<0$ & $y>0$

$y\le (x+1{)}^2$

$3^{rd}$ qnad

$x<0$ & $y<0$

$-y\le (x+1{)}^2$

$4^{th}$ quad

$y<0$ & $x>0$

$-y\le (x-1{)}^2$

Nature of $y=(x-1{)}^2$ is Parabola of type $x^2=4ay$

Required set of points including Dark line/boundary