Find the equation of the set of points such that the sum of its distances from (0, 3) and (0, -3) is 8.

Explanation

Let `P(x, y)` be any general point and let ` A-=(0, 3) and B-=(0,-3)` such that
`" "PA+PB=8`.
`rArr" "sqrt((x-0)^(2)+(y-3)^(2))+sqrt((x-0)^(2)+(y+3)^(2))=8`
`rArr" "sqrt(x^(2)+y ^(2)-6y+9)=8-sqrt(x^(2)+y^(2)+6y+9)`
Squaring both sides, we have
`rArr" "x^(2)+y^(2)-6y+9=64+x^(2)+y^(2)+6y+9- 16sqrt(x^(2)+y^(2)+6y+9)`
`rArr" "12y+64 =16sqrt(x^(2)+y^(2) +6y+9)`
` rArr" "3y+16=4sqrt(x^(2)+y^(2)+6y+9)`
Squaring again, we have
`therefore" "9y^(2)+256+96y=16(x^(2)+y^(2)+6y+9)`
`rArr" "16x^(2)+ 7y^(2)=112" "or" " (x^(2))/(7)+( y^(2))/(9)=1`
which is the required equation.