Question

If the line y = mx does not intersect the circle
$(x+10{)}^2+(y+10{)}^2=180$
then write the set of values taken by m

Answer 1

$y=mx$ does not intersect the circle

$(x+10{)}^2+(y+10{)}^2=180$
Now, $(x+10{)}^2+(mx+10{)}^2=180x^2+100+20x+m^2{x}^2+100+20mx=180x^2+m^2{x}^2+20mx+20x+20=0x^2(1+m^2)+20x(m+1)+20=0$
Since y = mx does not intersect so
$b^2-4ac<0\left(20\right(m+1){)}^2-4(1+m{)}^2\left(20\right)<0400(m+1{)}^2-80(1+m^2)<05m^2+5+10m-1-m^2<04m^2+10m+4<04m^2+8m+2m+4<04m(m+2)+2(m+2)<0(m+2\left)\right(4m+2)<0\Rightarrow m+2<0and4m+2>0m>-2andm<-\frac{1}{2}$
Its not possible
$\Rightarrow m+2>0and4m+2<0\Rightarrow m>-2andm<-\frac{1}{2}\Rightarrow -2<m<-\frac{1}{2}\Rightarrow mϵ(-2,-\frac{1}{2})$