To find the locus of a point \( P(x, y) \) which moves such that its distance from the point \((0, 3)\) is equal to the ordinate (y-coordinate) of \(P\), follow these steps:
1. Understand the conditions given:
- The point \( P \) has coordinates \( (x, y) \).
- The fixed point is \( (0, 3) \).
- The distance between \( P \) and \( (0, 3) \) is given to be equal to the ordinate of \( P \).
2. Express the distance between \( P \) and \( (0, 3) \) using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x - 0)^2 + (y - 3)^2}
\][/tex]
Which simplifies to:
[tex]\[
\text{Distance} = \sqrt{x^2 + (y - 3)^2}
\][/tex]
3. Set up the equation using the given condition:
According to the problem, this distance is equal to the ordinate (y-coordinate) of \( P \):
[tex]\[
\sqrt{x^2 + (y - 3)^2} = y
\][/tex]
4. Form the equation based on the condition:
[tex]\[
y = \sqrt{x^2 + (y - 3)^2}
\][/tex]
5. This equation represents the locus of the point \( P \):
[tex]\[
\boxed{y = \sqrt{x^2 + (y - 3)^2}}
\][/tex]
This equation provides the relationship between the coordinates [tex]\( x \)[/tex] and [tex]\( y \)[/tex] which the point [tex]\( P \)[/tex] must satisfy as it moves to keep the distance to [tex]\((0, 3)\)[/tex] equal to its ordinate [tex]\( y \)[/tex].
