A Focal Chord is a line perpendicular to the principal axis at the focus point. The focal chord is called the “primary foci” or “focal line.” It is a line that is used in geometrical optics. The distance between the two points of the focal chord is called the “focal distance” or “focal length.”

In a conic section focal chord is the chord passing through the focus of the conic section.

What is a Focal Cord?

A Focal chord is a chord of a conic section (parabola, ellipse, or hyperbola) that passes through one of its foci. In simpler terms, it is a line segment whose endpoints lie on the conic section and which also intersects the focus of the conic.

Focal Cord in Parabola

In a parabola, the focus is a fixed point, and the directrix is a fixed-line. A focal chord of a parabola is a line segment that passes through the focus and intersects the parabola at two points.

Standard Equation of a Parabola

Let’s consider the standard equation of a parabola: [Tex]y^ 2 =4ax[/Tex], Here, (a,0) is the focus of the parabola.

The product of the slopes of the lines connecting these points to the focus (a,0)(a, 0)(a,0) is -1.

Focal Cord in Ellipse

In an ellipse, a focal chord passes through one of the foci and intersects the ellipse at two points. For an ellipse centered at the origin with semi-major axis a and semi-minor axis b, the equation is x²/a²+y²/b²=1.

A focal cord in an ellipse is a line segment that:

• Connects two points on the ellipse.
• Passes through one of the foci of the ellipse.

For an ellipse with semi-major axis a and semi-minor axis b, the length of the focal cord passing through a focus can be calculated using the equation:

Length of Focal Cord: [Tex]\frac{2b^2}{a} [/Tex]

Factors that Help You to Determine the Focal Chord of Ellipse

However, a few factors can help you determine the focal chord of an ellipse.

• One important factor is the height of the ellipse. The higher the height, the longer the focal chord will be.
• Another important factor is the width of the ellipse. The wider the ellipse, the shorter the focal chord will be.
• One way to measure the focal chord of an ellipse is by using the height and width of the ellipse.

Focal Cord in Hyperbola

In a hyperbola, a focal chord passes through one of the foci and intersects the hyperbola at two points. For a hyperbola centered at the origin, the equation is (x²/a²)−(y²/b² ) =1.

The concept of a focal cord is similar but with some variations:

• Hyperbola has two foci, and a focal cord is a line segment that connects two points on the hyperbola and passes through one of these foci.

Properties:

• Length of a focal cord in a hyperbola is related to the transverse axis and the distance between the foci.
• Unlike ellipses, focal cords in hyperbolas are not necessarily perpendicular to any particular axis.

Properties of Focal Chord

Equation of the Chord: The chord passes through the focus (a,0). Let’s denote the endpoints of the chord as P( x₁,y₁ ) and Q(x₂,y₂)​. Since both points lie on the parabola:

[Tex]y_1^2 = 4ax_1[/Tex]

[Tex]y_2^2 = 4ax_2[/Tex]

Slope of Chord: The slope ? of the line passing through P and Q is:

[Tex]m = \frac{y_2 – y_1}{x_2 – x_1}[/Tex]

Length of Focal Chord: The length of the focal chord can be derived from the properties of the parabola and is given by: [Tex]PQ = \frac{2a}{|m|}[/Tex], where ? is the slope of the focal chord.

Length of Focal Chord

A focal chord of a parabola is a line segment that passes through the focus of the parabola and has its endpoints on the parabola itself. The latus rectum is the most well-known focal chord, being perpendicular to the parabola’s axis of symmetry.

Formula for Length of Focal Chord

For a parabola represented by the equation y²=4axy, the length L of a focal chord passing through the focus is given by:

[Tex]L = \frac{4a}{\cos^2 \theta}[/Tex]​

where θ is the angle between the focal chord and the axis of the parabola.

Example: Find the length of the focal chord of the parabola y² = 4ax which passes through the point (2a, 2a).

Solution:

For the parabola y² = 4ax

Equation of the focal chord passing through the focus (a,0) and point (2a,2a) can be derived by using the point-slope form of the line equation.

First, find the slope m of the chord:

[Tex]m = \frac{2a – 0}{2a – a} = \frac{2a}{a} = 2[/Tex]

So, the equation of the chord is:

⇒ y−0=2(x−a)

⇒ y=2x−2a

To find the points where this chord intersects the parabola y²=4ax:

(2x−2a)²=4ax

⇒4x² – 8ax + 4a² = 4ax

⇒4x²−12ax+4a²=0

⇒x²−3ax+a²=0

Solving for x using the quadratic formula:

[Tex]x = \frac{3a \pm \sqrt{9a^2 – 4a^2}}{2}[/Tex]

[Tex]x = \frac{3a \pm \sqrt{5a^2}}{2}[/Tex]}

[Tex]x = \frac{3a \pm a\sqrt{5}}{2}[/Tex]​​

Thus, the points of intersection are:

[Tex]x_1 = \frac{3a + a\sqrt{5}}{2}, \quad x_2 = \frac{3a – a\sqrt{5}}{2}[/Tex]

The corresponding y coordinates are:

[Tex]y_1 = 2\left(\frac{3a + a\sqrt{5}}{2}\right) – 2a = 3a + a\sqrt{5} – 2a = a + a\sqrt{5}[/Tex]

[Tex]y_2 = 2\left(\frac{3a – a\sqrt{5}}{2}\right) – 2a = 3a – a\sqrt{5} – 2a = a – a\sqrt{5}[/Tex]

Now, the length of the focal chord is given by the distance between these points:

[Tex]\text{Length} = \sqrt{\left(\frac{3a + a\sqrt{5}}{2} – \frac{3a – a\sqrt{5}}{2}\right)^2 + \left((a + a\sqrt{5}) – (a – a\sqrt{5})\right)^2}[/Tex]

[Tex]\text{Length} = \sqrt{\left(\frac{2a\sqrt{5}}{2}\right)^2 + \left(2a\sqrt{5}\right)^2}[/Tex]

[Tex]\text{Length} = \sqrt{(a\sqrt{5})^2 + (2a\sqrt{5})^2}[/Tex]

[Tex]\text{Length} = \sqrt{5a^2 + 20a^2}[/Tex]

[Tex]\text{Length} = \sqrt{25a^2}[/Tex]

[Tex]\text{Length} = 5a[/Tex]

So, the length of the focal chord is 5a.

Example on Focal Chord

Example 1: Consider an ellipse with a semi-major axis a = 5 and a semi-minor axis b = 3. The length of the focal cord through one of its foci is:

Solution:

Length of the focal chord is given by the formula:

Length (Focal Chord) = 2b²/a

= (2×3²)/5

= 18/5 = 3.6

Example 2: For a hyperbola with semi-major axis a = 4 and semi-minor axis b = 3, the length of the focal cord is:

Solution:

Length of the focal chord is given by the formula:

Length (Focal Chord) = 2b²/a

= (2×3²)/4

= 18/4 = 4.5

Example 3: For the parabola y² = 6x , find the coordinates of the points where a focal chord intersects the parabola and determine its length.

Solution:

For parabola y² = 6x

Length of focal chord is 4a = (4 ×3)/2 = 6

Importance of Focal Chord and Focal Distance

Focal chords have several significant properties and applications in geometry and physics:

• Geometric Properties: Understanding the properties of focal chords helps in solving complex geometric problems involving conic sections.
• Optics: In optics, focal chords are related to the paths of light rays and their reflection properties in parabolic mirrors and lenses.
• Engineering Applications: Focal chords play a role in the design of various engineering structures and components, particularly those involving parabolic and elliptical shapes.

Practice Question on Focal Chord

Q1. Given the parabola y² = 4ax find the length of the focal chord that passes through the point [Tex](4a, 4\sqrt{a})[/Tex].

Q2. Given the parabola y² = 8x calculate the length of the focal chord.

Q3. An ellipse has semi-major axis a=5 and semi-minor axis b=3. Find the length of the focal chord that passes through the focus.

Q4. Given the ellipse 4x²+9y²=36, calculate the length of the focal chord passing through one of its foci.

Q5. A hyperbola has semi-major axis a=4 and semi-minor axis b=3. Find the length of the focal chord that passes through the focus.

Q6. Given the hyperbola 9x²−16y²=144, calculate the length of the focal chord passing through one of its foci.

Conclusion

The focal cord is a fundamental concept in the study of conic sections, particularly ellipses and hyperbolas. It gives information about the geometric qualities and relationships of these shapes. Understanding the focal cord allows you to develop a better appreciation for the mathematical beauty and symmetry seen in conic sections.

Related Articles:

• Graph, properties, examples, and equation of parabola
• Conic Section

FAQs on Focal Chord

What is a focal cord?

A focal cord is a line segment that connects two points on a conic section (ellipse or hyperbola) and passes through one of its foci. It is an important geometric feature used to analyze and understand the properties of these shapes.

What is the significance of a focal cord in conic sections?

The focal cord is significant because it provides insight into the geometric properties of conic sections. It helps in understanding how distances and angles relate to the foci of the ellipse or hyperbola.

Are there any special properties of the focal cord in an ellipse?

Yes, in an ellipse, the focal cord passing through a focus is always perpendicular to the major axis when the endpoints lie on the ellipse. This property is useful in various geometric and algebraic calculations.