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Find the locus of a point [tex]\( P \)[/tex] which moves such that its distance from [tex]\((0, 3)\)[/tex] is equal to the ordinate of [tex]\( P \)[/tex].

Sagot :

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].
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