DIRECTRIX

Directrix

Definition Of Directrix Of A Conic Section

Directrix of a conic section is a line such that ratio of the distance of the points on the conic section from focus to its distance from directrix is constant.

More About Directrix of a Conic Section

If the ratio (r) of the distance of any point on the conic section from focus to its distance from directrix is equal to 1, then the conic section is a parabola. If it is less than 1, then it is ellipse; and if it is greater than 1, then the conic section is a hyperbola

Example Of Directrix Of A Conic Section

The given figure represents the directrix of Ellipse, Parabola, and Hyperbola

example of Directrix

The directrixes of three different conic sections, 'Ellipse', 'Parabola', and 'Hyperbola' are shown above.

Video Examples: Parabola Focus and Directrix 1

Solved Example on Directrix of a Conic Section

Ques: Find the equation of directrix of the parabola y² = 12x.

Choices:

A. y = - 3
B. x= 3
C. x = - 3
D. y = 3
E. X = 12
Correct Answer: C

Solution:

Step 1: y²= 12x [Write the equation of the parabola.]
Step 2: 4 P = 12, so P = 3 [Compare with y² = 4Px.]
Step 3: So, x = - 3 is the equation of the directrix of the parabola. [The equation of the directrix of the parabola y² = 4Px is x = - p.]

Quick Summary

The directrix is a line associated with conic sections (parabola, ellipse, hyperbola).
The ratio 'r' (eccentricity) determines the type of conic section: r=1 (parabola), r<1 (ellipse), r>1 (hyperbola).
The directrix helps define the shape and properties of the conic section.

\[ r = \frac{\text{distance to focus}}{\text{distance to directrix}} \]

🍎 Teacher Insights

Use visual aids and interactive software to demonstrate how changing the position of the directrix affects the shape of the conic section. Emphasize the eccentricity as a defining characteristic.

🎓 Prerequisites

Algebra
Coordinate Geometry
Parabola
Ellipse
Hyperbola
Focus

Check Your Knowledge

Q1: The equation of a parabola is y^2 = 4x. What is the equation of its directrix?

A. x = 1 B. x = -1 C. y = 1 D. y = -1

Q2: If the eccentricity of a conic section is 0.5, what type of conic section is it?

A. Parabola B. Ellipse C. Hyperbola D. Circle

Frequently Asked Questions

Q: What is the relationship between the focus and directrix of a parabola?
A: For a parabola, every point on the curve is equidistant from the focus and the directrix.

Q: Does every conic section have a directrix?
A: Yes, all conic sections (ellipse, parabola, and hyperbola) have at least one directrix.