Find the locus of point P such that the sum of its distances from (0,2) and (0,-2) is 6.
My attempt: I applied the distance formula but it's not helping. The locus problems always trouble me. Could you tell me a general approach to solve locus problems? And how should one go about such problems using just geometry without the equations of coordinate geometry?
On
The locus of a point $P$ whose sum of distances from two points is constant, is known as an ellipse. The two points are called the foci. One focus, another focus, two foci. (Latin). Suppose the foci are $(a,b)$ and $(c,d)$. Then writing down the geometrical condition gives:
$$\sqrt{(x-a)^2+(y-b)^2}+\sqrt{(x-c)^2+(y-d)^2}=\hbox{const}$$
In your case, $a=0,b=2,c=0,d=-2$ and the constant is $6$.
In general, there is no one method to solve locus problems without understanding what are the equations that come with the geometry.
HINT: get the equation $$\sqrt{x^2+(y-2)^2}+\sqrt{x^2+(y+2)^2}=6$$