Equation of a Hyperbola: Exploring Distances and Coordinates

When dealing with the problem of finding an equation that satisfies the conditions of a point's distances from two fixed points, we delve into the geometric properties of hyperbolas. This article outlines how to derive the equation for such a hyperbola and explores the underlying principles.

Introduction to the Problem

The problem presented is to find an equation that must be satisfied by the coordinates of any point ( (x, y) ) whose distance from the point ( (5, 3) ) is always two units greater than its distance from the point ( (-4, -2) ).

Deriving the Equation

Let's denote the distances as follows:

1. The distance from ( (x, y) ) to the point ( (5, 3) ): (sqrt{(x - 5)^2 (y - 3)^2})

2. The distance from ( (x, y) ) to the point ( (-4, -2) ): (sqrt{(x 4)^2 (y 2)^2})

According to the problem, the first distance is always two units greater than the second distance:

[sqrt{(x - 5)^2 (y - 3)^2} sqrt{(x 4)^2 (y 2)^2} 2]

To simplify, let's isolate the square roots and then square both sides:

[(x - 5)^2 (y - 3)^2 left[ sqrt{(x 4)^2 (y 2)^2} 2 right]^2]

Expanding both sides, we get:

[(x - 5)^2 (y - 3)^2 (x 4)^2 (y 2)^2 4sqrt{(x 4)^2 (y 2)^2} 4]

Simplifying further, we obtain:

[x^2 - 1 25 y^2 - 6y 9 x^2 8x 16 y^2 4y 4 4sqrt{(x 4)^2 (y 2)^2} 4]

Combining like terms, we get:

[-18x - 10y 20 4sqrt{(x 4)^2 (y 2)^2}]

Squaring both sides again to eliminate the square root:

[324x^2 36 100 36y 200y 100y^2 400 16(x^2 8x 16 y^2 4y 4)]

Further simplification yields:

[324x^2 36 100 36y 200y 100y^2 400 16x^2 128x 256 16y^2 64y 64]

Simplifying the entire equation:

[308x^2 36 100y^2 36y 136y - 116 0]

This is a hyperbola equation that satisfies the given condition.

The Geometric Interpretation

This hyperbola has two branches, each corresponding to the points where the difference in distances is constant. The fixed points ( (5, 3) ) and ( (-4, -2) ) are the foci of the hyperbola. The locus of points ( (x, y) ) that satisfies the equation is a branch of a hyperbola where the difference in distances to the foci is 2 units.

Exploring the Symmetry and Asymptotes

The hyperbola is symmetric about the x-axis and y-axis. The center of the hyperbola is not at the origin but is shifted. By shifting the foci to the origin, we can derive the standard form of the hyperbola equation:

[ sqrt{(x - 5)^2 (y - 3)^2} sqrt{(x 4)^2 (y 2)^2} 2 ]

By simplifying and comparing to the standard form of a hyperbola, we can derive the equation:

[ frac{(x - h)^2}{a^2} - frac{(y - k)^2}{b^2} 1 ]

Here, ( (h, k) ) is the center, and ( a ), ( b ) are the semi-major and semi-minor axes.

Conclusion

The problem of finding a point's coordinates whose distance to one fixed point is always 2 units greater than the distance to another fixed point can be solved by deriving a hyperbola equation. Understanding the geometric properties and the derived equation allows for a deeper exploration of coordinate geometry and the properties of hyperbolas.

Keywords

equation of a hyperbola, distance equation, coordinate geometry