Understanding Focal Distance

Focal distance is a concept that arises in the study of conic sections, which include ellipses, hyperbolas, and parabolas. In the context of hyperbolas, the focal distance refers to the distance from a point on the hyperbola to one of the foci of the hyperbola.

Hyperbola

A hyperbola is a type of conic section that can be defined as the set of all points (x, y) in the plane such that the absolute difference of the distances from (x, y) to two fixed points (the foci) is constant. This constant is usually denoted as 2a, where a is the distance from the center of the hyperbola to a vertex on the hyperbola.

Standard Equation of a Hyperbola

The standard equation of a hyperbola centered at the origin with its transverse axis along the x-axis is:

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

Where:

• a is the distance from the center to a vertex on the hyperbola.
• b is the distance from the center to a co-vertex on the hyperbola.

The foci are located at (±c, 0) where c = \sqrt{a^2 + b^2}.

Focal Distance

The focal distance of a point P(x, y) on the hyperbola to a focus F is the distance between P and F. For a hyperbola, there are two focal distances for each point on the hyperbola, one for each focus.

Formula for Focal Distance

For a point P(x, y) on the hyperbola, the focal distances to the foci F1(-c, 0) and F2(c, 0) are given by:

$$ d_1 = \sqrt{(x + c)^2 + y^2} $$

$$ d_2 = \sqrt{(x - c)^2 + y^2} $$

Where:

• d_1 is the distance to the focus F1.
• d_2 is the distance to the focus F2.

Properties of Focal Distances

For a hyperbola, the difference between the focal distances to any point on the hyperbola is constant and equal to 2a.

$$ |d_1 - d_2| = 2a $$

Table of Differences and Important Points

Property	Ellipse	Hyperbola	Parabola
Definition	Sum of distances to foci is constant	Difference of distances to foci is constant	Distance to focus equals distance to directrix
Standard Equation	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$y^2 = 4ax$
Foci	(±c, 0) or (0, ±c)	(±c, 0) or (0, ±c)	(a, 0) or (0, a)
Focal Distance	d = \sqrt{(x \pm c)^2 + y^2}	d_1 = \sqrt{(x + c)^2 + y^2}, d_2 = \sqrt{(x - c)^2 + y^2}	`d =

Examples

Example 1: Hyperbola with Given Point

Given a hyperbola with equation $\frac{x^2}{16} - \frac{y^2}{9} = 1$ and a point P(5, 0) on the hyperbola, find the focal distances.

Solution:

First, identify a and b:

• a^2 = 16 so a = 4
• b^2 = 9 so b = 3

Calculate c:

• c = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Now, use the formulas for d_1 and d_2:

• d_1 = \sqrt{(5 + 5)^2 + 0^2} = \sqrt{100} = 10
• d_2 = \sqrt{(5 - 5)^2 + 0^2} = \sqrt{0} = 0

The focal distances are 10 and 0.

Example 2: Verifying the Property

Verify that the difference between the focal distances for the point P(5, 0) on the hyperbola from Example 1 is equal to 2a.

Solution:

We already know d_1 = 10 and d_2 = 0, and a = 4.

Calculate the difference:

• |d_1 - d_2| = |10 - 0| = 10

Since 2a = 2 * 4 = 8, the property does not seem to hold. However, we must remember that the point P(5, 0) is not actually on the hyperbola, as it does not satisfy the hyperbola's equation. This is why the property does not hold in this case. If P were a point on the hyperbola, the property would indeed hold.

In conclusion, the focal distance in hyperbolas is a fundamental concept that helps define the shape and properties of the curve. Understanding the focal distances and their properties is essential for solving problems related to hyperbolas in mathematics.