Position of a Point w.r.t. Hyperbola

Understanding the position of a point with respect to a hyperbola is crucial in coordinate geometry, particularly in the context of conic sections. A hyperbola is a set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant. The standard equation of a hyperbola centered at the origin with its transverse axis along the x-axis is given by:

$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$

Where $a$ is the distance from the center to the vertices on the x-axis, and $b$ is the distance from the center to the vertices on the y-axis.

Determining the Position of a Point

To determine the position of a point $(x_1, y_1)$ with respect to a hyperbola, we substitute the coordinates of the point into the hyperbola's equation. The resulting expression helps us to conclude whether the point lies inside, on, or outside the hyperbola.

The general procedure is as follows:

1. Substitute $(x_1, y_1)$ into the hyperbola's equation.
2. Evaluate the expression.
3. Based on the result, determine the position of the point.

The table below summarizes the possible outcomes and their interpretations:

Result of Substitution	Interpretation	Position of Point
Equals 1	Point lies on the hyperbola	On the hyperbola
Less than 1	Point lies inside the hyperbola	Inside the hyperbola
Greater than 1	Point lies outside the hyperbola	Outside the hyperbola

Formulas and Examples

Let's consider the standard equation of a hyperbola:

$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$

For a point $(x_1, y_1)$, we substitute it into the equation:

$$ \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} - 1 = S $$

Where $S$ is the result of the substitution.

Example 1: Point on the Hyperbola

Let's say we have a hyperbola with $a = 2$ and $b = 3$, and we want to determine the position of the point $(4, 3)$.

Substitute $(4, 3)$ into the hyperbola's equation:

$$ \frac{4^2}{2^2} - \frac{3^2}{3^2} = \frac{16}{4} - \frac{9}{9} = 4 - 1 = 3 $$

Since $3 > 1$, the point $(4, 3)$ lies outside the hyperbola.

Example 2: Point Inside the Hyperbola

Consider the same hyperbola, and let's determine the position of the point $(1, 1)$.

Substitute $(1, 1)$ into the hyperbola's equation:

$$ \frac{1^2}{2^2} - \frac{1^2}{3^2} = \frac{1}{4} - \frac{1}{9} = \frac{9 - 4}{36} = \frac{5}{36} $$

Since $\frac{5}{36} < 1$, the point $(1, 1)$ lies inside the hyperbola.

Example 3: Point on the Hyperbola

Now, let's check the position of the point $(2\sqrt{2}, 3)$.

Substitute $(2\sqrt{2}, 3)$ into the hyperbola's equation:

$$ \frac{(2\sqrt{2})^2}{2^2} - \frac{3^2}{3^2} = \frac{8}{4} - \frac{9}{9} = 2 - 1 = 1 $$

Since the result is exactly 1, the point $(2\sqrt{2}, 3)$ lies on the hyperbola.

Conclusion

The position of a point relative to a hyperbola can be determined by substituting the point's coordinates into the hyperbola's equation. The result of this substitution reveals whether the point is inside, on, or outside the hyperbola. This concept is fundamental in the study of conic sections and has applications in various fields, including physics, engineering, and computer graphics.