Understanding the Points A and B
The points given are A(0, 4) and B(0, -4). These points lie on the y-axis, with A above the origin and B below it.
Definition of Locus
The locus of a point P(x, y) is the set of all points that satisfy a certain condition. In this case, we need to find the locus such that the difference in distances from point P to points A and B equals 6.
Distance Formula
Using the distance formula, the distances from point P to points A and B can be computed as follows:
- Distance AP = √((x - 0)² + (y - 4)²)
- Distance BP = √((x - 0)² + (y + 4)²)
Setting Up the Equation
According to the problem, the condition is:
AP - BP = 6
This can be rewritten using the distances we just calculated:
√((x - 0)² + (y - 4)²) - √((x - 0)² + (y + 4)²) = 6
Squaring the Equation
To eliminate the square roots, square both sides of the equation:
((x - 0)² + (y - 4)²) - 2√(((x - 0)² + (y - 4)²)((x - 0)² + (y + 4)²)) + ((x - 0)² + (y + 4)²) = 36
This will yield an equation that can be simplified.
Final Form of the Locus
The result will be a hyperbola, centered along the y-axis, opening vertically. The locus of point P will represent all points equidistant from A and B, differing by 6 units in distance.
In conclusion, the locus of the point P is a hyperbola characterized by the condition AP - BP = 6, illustrating the geometric relationship between points A, B, and any point P in the plane.