Question

Find the locus of a point which moves such that the sum of its distance from $(3,0)$ and $(-3,0)$ is less then $9$.

Answer 1

Let the point be $(h,k)$ and $A(3,0)$ and $B(-3,0)$

$AC+AC<9$

$=>\sqrt{{(h-3)}^2+k^2}+\sqrt{{(h+3)}^2+k^2}<9=>{(h-3)}^2+k^2<81+{(h+3)}^2+k^2-18\sqrt{{(h+3)}^2+k^2}=>{\left(\frac{-6h-81}{18}\right)}^2<{(h+3)}^2+k^2=>\frac{h^2}{9}+\frac{81}{4}+\frac{27}{2}h<h^2+3h+9+k^2=>\frac{8h^2}{9}-\frac{21h}{2}+k^2-\frac{45}{4}>0=>32h^2-378h+36k^2-405>0$

locus $=>32h^2-378h+36k^2-405>0$