Question

the point (x,y) is more than 2 units but less than 6 units from the center (-1,3) of a circle. Can you describe the graph of these points?

Answer 1

Task:

The point $(x,y)$ is more than 2 units but less than 6 units from the center $(-1,3)$ of a circle. Can you describe the graph of these points?

Solution:

The point $(x,y)$ is more than 2 units means a circle with radius of 2(internal circle).

The point $(x,y)$ is more than 6 units means a circle with radius of 6(external circle).

So we get equations:

$$\left\{ \begin{array}{l} (x + 1) ^ {2} + (x - 3) ^ {2} > 2 ^ {2} \\ (x + 1) ^ {2} + (x - 3) ^ {2} < 6 ^ {2} \end{array} \right.$$ $$\left\{ \begin{array}{l} (x + 1) ^ {2} + (x - 3) ^ {2} > 4 \\ (x + 1) ^ {2} + (x - 3) ^ {2} < 3 6 \end{array} \right.$$

Plot it:

The graph of these points is region bounded by two concentric circles radius of 2 and 6 with center in $(-1,3)$ , while these two circles are not included in this area.

Answer: an annulus without outer circles.