Step-by-Step Method to Locate the Directrix and Focus of a Parabola
Directrix Of Parabola matters in school and competitive exams because it helps students determine and analyze the geometric properties of parabolas. Mastering this concept allows you to construct equations, locate focus points, and solve real-world geometry problems with ease, making it valuable for both academics and practical applications.
Formula Used in Directrix Of Parabola
The standard formula is:
For \( y^2 = 4ax \), the directrix is \( x = -a \);
For \( x^2 = 4ay \), the directrix is \( y = -a \).
For general parabolas with vertex \((h, k)\):
\( (y - k)^2 = 4a(x - h) \) → directrix: \( x = h - a \)
\( (x - h)^2 = 4a(y - k) \) → directrix: \( y = k - a \)
Here’s a helpful table to understand directrix of parabola more clearly:
Directrix Of Parabola Table
| Standard Parabola Equation | Axis | Directrix Equation |
|---|---|---|
| \( y^2 = 4ax \) | x-axis | \( x = -a \) |
| \( y^2 = -4ax \) | x-axis | \( x = a \) |
| \( x^2 = 4ay \) | y-axis | \( y = -a \) |
| \( x^2 = -4ay \) | y-axis | \( y = a \) |
This table shows how the equation of the directrix of parabola depends on the orientation and standard form of the given parabola.
Worked Example – Solving a Problem
Let’s solve a directrix of parabola question step by step:
1. The equation of the parabola is \( (y - 2)^2 = 8(x + 1) \).2. Compare with the standard form \( (y - k)^2 = 4a(x - h) \). Here, \( h = -1 \), \( k = 2 \), \( 4a = 8 \) → \( a = 2 \).
3. Directrix equation for this form: \( x = h - a \)
4. Substitute: \( x = -1 - 2 = -3 \).
5. Final answer: The directrix is \( x = -3 \).
Understanding these steps is crucial for exams like JEE, boards, and olympiads. For more on parabola forms, see Equation of Parabola and Standard Equation of Parabola 404.
Practice Problems
- Find the directrix of the parabola \( y^2 = 12x \).
- Given \( x^2 = -16y \), what is the directrix?
- If the vertex is at (2, 3) and equation is \( (x-2)^2 = 8(y-3) \), determine the directrix.
- Write the directrix for \( (y+1)^2 = -20(x-4) \).
Common Mistakes to Avoid
- Swapping the sign when copying the directrix formula (e.g., writing \( x = a \) instead of \( x = -a \)).
- Confusing the directrix line equation with the focus point coordinates.
- Not matching the equation’s orientation before applying the formula.
Real-World Applications
Directrix of parabola concepts are crucial in designing satellite dishes, car headlights, and even bridges. They help solve engineering, design, and physics problems. With Vedantu, students learn how directrix truly connects maths to everyday innovations. For more properties, visit Properties of Parabola.
We explored the idea of Directrix Of Parabola, how to use its formulae, solve step-wise problems, and apply it in real world. Practice more with Vedantu and deepen your understanding alongside related concepts like conic sections, and parabola graph.
FAQs on What Is the Directrix of a Parabola and How Do You Find It?
1. How do you find the directrix of a parabola?
To find the directrix of a parabola, first identify its standard equation. For a parabola with equation y2 = 4ax, the directrix is given by the formula x = -a. For x2 = 4ay, the directrix is y = -a. Always compare your equation to a standard form to determine the value of a, then use it in the directrix formula.
2. What does a directrix look like?
The directrix of a parabola is a straight line that lies outside the curve. It is always perpendicular to the axis of symmetry of the parabola and is not the same as the parabola itself. On a graph, it appears as a dashed or solid line parallel to the axis of the parabola, helping define its geometric properties.
3. Is the directrix always opposite of focus?
Yes, the directrix is always located opposite to the focus relative to the vertex of the parabola. The vertex lies exactly halfway between the focus and the directrix, demonstrating the parabola's reflective property.
4. Are vertex and directrix the same?
No, the vertex and directrix are not the same. The vertex is the point where the parabola changes direction, while the directrix is a fixed straight line used in the parabola's definition. The vertex is always equidistant from the focus and the directrix.
5. What is the formula for the directrix of the parabola y2 = 4ax?
For the standard parabola y2 = 4ax, the directrix is the line x = -a. Here, 'a' is the distance from the vertex to the focus, and the directrix is vertical and to the left of the vertex if a > 0.
6. What is the directrix of the parabola x2 = -4ay?
For the equation x2 = -4ay, the directrix is y = a. Here, the parabola opens downwards, and the directrix is a horizontal line above the vertex.
7. How do you find the focus of a parabola?
To find the focus of a parabola, use its standard equation. For y2 = 4ax, the focus is at (a, 0). For x2 = 4ay, it is at (0, a). The parameter 'a' determines how far the focus is from the vertex along the axis of symmetry.
8. What is the distance between the focus and the directrix of a parabola?
The distance between the focus and the directrix is equal to 2a for a parabola of form y2 = 4ax or x2 = 4ay. Here 'a' is the distance from the vertex to either the focus or directrix.
9. What is the definition of the directrix of a parabola?
The directrix of a parabola is a fixed straight line such that every point on the parabola is the same distance from the focus as it is from the directrix. This property defines the parabola geometrically.
10. How can you graph a parabola and its directrix?
To graph a parabola and its directrix:
• Write the equation in standard form (e.g., y2 = 4ax or x2 = 4ay)
• Plot the vertex (usually at the origin for standard form)
• Plot the focus at (a, 0) or (0, a)
• Draw the directrix as the line x = -a or y = -a
• Sketch the parabola as the set of points equidistant from the focus and directrix
11. What is a directrix of a parabola in Hindi?
परबोला की डायरेक्ट्रिक्स वह स्थिर रेखा है जिससे परबोला पर स्थित किसी बिंदु की दूरी, उसी बिंदु से फोकस तक की दूरी के बराबर होती है। यह परबोला की ज्यामितीय परिभाषा है।
12. Can I use a calculator to find the directrix of a parabola?
Yes, you can use an online calculator or mathematical tools to quickly find the directrix by entering the standard form of the parabola's equation. The calculator uses the formula x = -a or y = -a depending on the parabola's orientation.
