To find the set of all points that are equidistant from the line [tex]\( y = 1 \)[/tex] and the point [tex]\((0, 3)\)[/tex], let's follow these steps:
1. Express the distance from a point [tex]\((x, y)\)[/tex] to the line [tex]\(y = 1\)[/tex]:
- For the line equation [tex]\(y - 1 = 0\)[/tex], the general formula for distance from a point [tex]\((x_1, y_1)\)[/tex] to a line [tex]\(Ax + By + C = 0\)[/tex] is given by:
[tex]\[
\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\][/tex]
- Here the line [tex]\(y - 1 = 0\)[/tex] implies [tex]\(A = 0\)[/tex], [tex]\(B = 1\)[/tex], and [tex]\(C = -1\)[/tex]. The distance from a point [tex]\((x, y)\)[/tex] to this line simplifies to:
[tex]\[
\text{Distance} = \frac{|0 \cdot x + 1 \cdot y - 1|}{\sqrt{0^2 + 1^2}} = |y - 1|
\][/tex]
2. Express the distance from a point [tex]\((x, y)\)[/tex] to the point [tex]\((0, 3)\)[/tex]:
- The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- For our points [tex]\((x, y)\)[/tex] and [tex]\((0, 3)\)[/tex], the distance is:
[tex]\[
\text{Distance} = \sqrt{(x - 0)^2 + (y - 3)^2} = \sqrt{x^2 + (y - 3)^2}
\][/tex]
3. Set the distances equal to each other:
- We are looking for the points where the distance from [tex]\((x, y)\)[/tex] to the line [tex]\(y = 1\)[/tex] is equal to the distance from [tex]\((x, y)\)[/tex] to the point [tex]\((0, 3)\)[/tex]:
[tex]\[
|y - 1| = \sqrt{x^2 + (y - 3)^2}
\][/tex]
Thus, the equation representing the set of all points that are equidistant from the line [tex]\(y = 1\)[/tex] and the point [tex]\((0, 3)\)[/tex] is:
[tex]\[ \boxed{\text{Eq} (|y - 1|, \sqrt{x^2 + (y - 3)^2})} \][/tex]
