Question

1. find a formula that expresses the fact that an arbitrary point p(x,y) is a distance 5 from the origin. describe the set of all such points.

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, one point is $P(x,y)$ and the other is the origin $(0,0)$, and $d = 5$.

Step2: Substitute values into formula

$5=\sqrt{(x - 0)^2+(y - 0)^2}$.

Step3: Simplify the equation

Squaring both sides to get rid of the square - root, we have $25=x^{2}+y^{2}$.
The set of all such points is a circle centered at the origin $(0,0)$ with a radius of $r = 5$.

Answer:

The formula is $x^{2}+y^{2}=25$, and the set of all such points is a circle centered at the origin with radius 5.