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).
$4⩽x^2+y^2⩽2(|x|+|y|)$

Answer 1

R.E.F image

For $x^2+y^2\le 2(|x|+|y|)$

We can write $x^2=|x{|}^2$ & $y^2=|y{|}^2$

Now,

$(x{)}^2-2|x|+|y{|}^2-2|y|\le 0$

add $+2$ & $-2$

$(x{)}^2-2|x|+1+|y{|}^2-2|y|+1-2\le 0$

$(|x|-1{)}^2+(|y|-1{)}^2-2|y|+1-2\le 0$

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

Here, $(x\pm 1{)}^2+(y\pm 1{)}^2=2$

represent a circle, so Graph will be circle

let's divide into quadrants

$I^{st}$ quadr

$x>0$ & $y>0$

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

$2^{nd}$ quadr

$x<0$ & $y>0$

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

$3^{rd}$ quadr

$x<0$ & $y<0$

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

$4^{th}$ quadr

$x>0$ & $y<0$

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