What Is the Directrix of a Parabola Formula Derivation and Solved Examples
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 Directrix of Parabola Explained with Definition and Formula
1. What is the directrix of a parabola?
The directrix of a parabola is a fixed straight line such that every point on the parabola is equidistant from the directrix and a fixed point called the focus. By definition of a parabola:
- Distance of any point on the parabola from the focus = distance from the directrix.
- The directrix is always perpendicular to the axis of symmetry.
- It lies outside the curve, opposite the focus.
2. What is the formula for the directrix of a parabola?
The formula of the directrix depends on the standard form of the parabola.
- For y² = 4ax, directrix is x = −a.
- For y² = −4ax, directrix is x = a.
- For x² = 4ay, directrix is y = −a.
- For x² = −4ay, directrix is y = a.
3. How do you find the directrix of a parabola?
To find the directrix of a parabola, first convert the equation into standard form and identify the value of a.
- Step 1: Write the equation in standard form like y² = 4ax or x² = 4ay.
- Step 2: Identify the value of a from 4a.
- Step 3: Use the corresponding directrix formula (e.g., x = −a or y = −a).
4. What is the directrix of y² = 4ax?
The directrix of y² = 4ax is x = −a. In this standard parabola:
- Vertex is at (0, 0).
- Focus is at (a, 0).
- Directrix is the vertical line x = −a.
5. What is the directrix of x² = 4ay?
The directrix of x² = 4ay is y = −a. For this upward-opening parabola:
- Vertex is at (0, 0).
- Focus is at (0, a).
- Directrix is the horizontal line y = −a.
6. What is the relationship between the focus and directrix of a parabola?
The focus and directrix of a parabola are related by the property that every point on the parabola is equidistant from both. Specifically:
- Distance from any point on the parabola to the focus equals its perpendicular distance to the directrix.
- The vertex lies midway between the focus and the directrix.
- The distance from vertex to focus equals the distance from vertex to directrix, both equal to a.
7. How do you find the directrix of a parabola in vertex form?
To find the directrix in vertex form, compare the equation with the standard shifted form and identify a.
- For (y − k)² = 4a(x − h), directrix is x = h − a.
- For (x − h)² = 4a(y − k), directrix is y = k − a.
8. What is the directrix of the parabola y² = 8x?
The directrix of y² = 8x is x = −2. Comparing with y² = 4ax:
- 4a = 8 ⇒ a = 2.
- Directrix = x = −a = −2.
- Focus = (2, 0).
9. Is the directrix always outside the parabola?
Yes, the directrix of a parabola always lies outside the curve. This happens because:
- The vertex lies between the focus and the directrix.
- The parabola opens toward the focus and away from the directrix.
- No point of the parabola lies on the directrix.
10. What is the difference between the focus and directrix of a parabola?
The focus is a fixed point, while the directrix is a fixed straight line used to define a parabola.
- Focus: A single point inside the curve.
- Directrix: A line outside the curve.
- Definition rule: Distance to focus = perpendicular distance to directrix.
