Focal Chord

A focal chord of a conic section (ellipse, parabola, or hyperbola) is a line segment that passes through the focus of the conic and has both endpoints on the curve. In the case of a hyperbola, there are two foci, and therefore a focal chord can be associated with either focus.

Properties of Focal Chords in Hyperbolas

A hyperbola is defined as the set of all points in the plane such that the absolute 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:

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

where $2a$ is the length of the transverse axis, and $2b$ is the length of the conjugate axis.

Important Points

• The foci of the hyperbola are at $(\pm c, 0)$, where $c = \sqrt{a^2 + b^2}$.
• The focal chord perpendicular to the transverse axis is called the latus rectum.
• The length of the latus rectum is given by $ \frac{2b^2}{a} $.

Focal Chord Length

For a focal chord $PQ$ of the hyperbola that passes through the focus $F_1(c, 0)$, if $P(x_1, y_1)$ and $Q(x_2, y_2)$ are the endpoints on the hyperbola, then the product of the abscissas of $P$ and $Q$ is constant and equal to $-a^2$:

$$ x_1 \cdot x_2 = -a^2 $$

This is because the distances $PF_1$ and $QF_1$ satisfy the hyperbola's definition, and by applying the distance formula and simplifying, we arrive at the above relationship.

Examples

Example 1: Latus Rectum

Find the length of the latus rectum of the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$.

Solution:

Given $a^2 = 16$ and $b^2 = 9$, we find $a = 4$ and $b = 3$. The length of the latus rectum is:

$$ L = \frac{2b^2}{a} = \frac{2 \cdot 9}{4} = \frac{18}{4} = 4.5 $$

Example 2: Focal Chord Length

For the hyperbola $\frac{x^2}{25} - \frac{y^2}{16} = 1$, find the length of a focal chord that passes through the focus at $(5, 0)$ and has one endpoint at $(5, 4)$.

Solution:

Let the other endpoint be $(x_2, y_2)$. Using the property of the focal chord, we have:

$$ x_1 \cdot x_2 = -a^2 $$

$$ 5 \cdot x_2 = -25 $$

$$ x_2 = -5 $$

Now, we need to find $y_2$. We use the equation of the hyperbola:

$$ \frac{x_2^2}{25} - \frac{y_2^2}{16} = 1 $$

$$ \frac{(-5)^2}{25} - \frac{y_2^2}{16} = 1 $$

$$ \frac{y_2^2}{16} = \frac{5}{25} - 1 $$

$$ y_2^2 = 16 \left( \frac{1}{5} - 1 \right) $$

$$ y_2^2 = 16 \left( -\frac{4}{5} \right) $$

$$ y_2 = \pm 4 \sqrt{\frac{4}{5}} $$

Since $y_2$ must be positive (as $y_1$ is positive), we choose the positive root:

$$ y_2 = 4 \sqrt{\frac{4}{5}} $$

The length of the focal chord is the distance between $(5, 4)$ and $(-5, 4 \sqrt{\frac{4}{5}})$, which can be found using the distance formula:

$$ L = \sqrt{(5 - (-5))^2 + (4 - 4 \sqrt{\frac{4}{5}})^2} $$

$$ L = \sqrt{(10)^2 + \left(4 - 4 \sqrt{\frac{4}{5}}\right)^2} $$

Differences and Important Points

Property	Parabola	Hyperbola
Definition	Set of points equidistant from a point (focus) and a line (directrix).	Set of points where the absolute difference of distances from two fixed points (foci) is constant.
Foci	One focus	Two foci
Focal Chord	Passes through the focus and has endpoints on the curve.	Passes through one of the foci and has endpoints on the curve.
Latus Rectum	Line segment perpendicular to the axis of symmetry and passing through the focus.	Focal chord perpendicular to the transverse axis.
Equation for Focal Chord	$x_1 \cdot x_2 = -a^2$ (for horizontal parabola)	$x_1 \cdot x_2 = -a^2$

In conclusion, the focal chord is an important concept in the study of conic sections, particularly hyperbolas. Understanding its properties and how to calculate its length is essential for solving problems related to hyperbolas in exams.