If $r = \langle x,y\rangle$, $r_1 = \langle x_1, y_1\rangle$, and $r_2 = \langle x_2,y_2\rangle$, describe the set of all points $(x,y)$ ...

Question

[2D - vector application] If $r = \langle x,y\rangle$, $r_1 = \langle x_1, y_1\rangle$, and $r_2 = \langle x_2,y_2\rangle$, describe the set of all points $(x,y)$ such that $|r - r_1| + |r - r_2| = k$, where $k > |r_1 - r_2|$.

Please help! $r$, $r_1$, and $r_2$ are all vectors. It wouldn't let me use angled brackets, and I don't know how to format stuffs... so I used double parentheses to indicate angled brackets.

FYI, this problem is from Stewart Calculus Section 12.2 Problem 42

Answer 1

Hints:

It seems to be you want

$$\sqrt{(x-x_1)^2+(y-y_1)^2}+\sqrt{(x-x_2)^2+(y-y_2)^2}>\sqrt{(x_1-x_2)^2+(y_1^2-y_2)^2}\iff$$

Now, remember the definition of

Ellipse: It is the locus of all points the sum of shose distances from two fixed points is a constant ...