Question

Find the equation of the locus of all points the sum of whose distance from $(3,0)$ and $(9,0)$ is $12$.

Answer 1

Distance between two points $(x_1,y_1),(x_2,y_2)$ is

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Let 5(3,0) and p(q,0) be two points. Let (x,4) be moving point

sum of distnce between them is 12.... (given)

$\therefore \sqrt{(x-3{)}^2+y^2}+\sqrt{(x-9{)}^2+y^2}=12$

$\sqrt{(x-9{)}^2+y^2}=12-\sqrt{(x-3{)}^2+y^2}$

squaring,

$x^2-18x+81+y^2=144-24\sqrt{(x-3{)}^2+y^2+x^2-6x+9+y^2}$

simplifying,

$12x+72=24\sqrt{(x-3{)}^2+y^2}$

$x+6=27\sqrt{(x-3{)}^2+y^2}$

squaring once again,

$(x+6{)}^2=4((x-3{)}^2+y^2)$

$x^2+12x+36=4[x^2-6x+9+y^2]$

$3x^2+4y^2-36x=0$

is the required equation of set of points.