How to Find the Focus Point and 4p Value of a Parabola
Focus of Parabola is a must-know for school boards and entrance exams, because it defines how a parabola is drawn and used in real problems—from mirrors to satellite dishes. Understanding the focus helps you solve graph-based questions and visualise geometric features with ease.
Formula Used in Focus of Parabola
The standard formula is: \( (x - h)^2 = 4a(y - k) \)
For this, the focus is at \( (h, k + a) \).
For \( (y - k)^2 = 4a(x - h) \), the focus is \( (h + a, k) \).
Here’s a helpful table to understand Focus of Parabola more clearly:
Focus of Parabola Table
| Equation | Axis | Vertex | Focus |
|---|---|---|---|
| \( y^2 = 4ax \) | x-axis | (0, 0) | (a, 0) |
| \( x^2 = 4ay \) | y-axis | (0, 0) | (0, a) |
| \( (x-h)^2 = 4a(y-k) \) | y = k | (h, k) | (h, k+a) |
| \( (y-k)^2 = 4a(x-h) \) | x = h | (h, k) | (h+a, k) |
This table helps you match the equation type to the correct position of the focus for any parabola – a skill needed for quick board and competitive exam problem solving. For more on how the equation links to the graph, see our Parabola Graph page.
How to Find the Focus of a Parabola – Step by Step
Find the focus in 4 steps:
1. Rearrange your equation into a standard form: \( (x-h)^2 = 4a(y-k) \) or \( (y-k)^2 = 4a(x-h) \).
2. Identify the vertex: It’s (h, k) from the equation.
3. Find the value of ‘a’: This is from the equation part \( 4a \), so \( a = \frac{\text{coefficient}}{4} \).
4. Use the correct formula for the focus: For \( (x-h)^2 = 4a(y-k) \), focus is (h, k+a). For \( (y-k)^2 = 4a(x-h) \), focus is (h+a, k).
You’ll use these steps in all types of focus questions, from basic to advanced. See Properties of Parabola for more properties linked to the focus.
Worked Example – Solving a Focus of Parabola Problem
Question: Find the focus of the parabola \( (x - 2)^2 = 12(y + 3) \).
1. Identify the equation: It matches \( (x-h)^2 = 4a(y - k) \) where h = 2, k = -3.
2. Find a: \( 4a = 12 \implies a = 3 \).
3. The vertex is (2, -3).
4. According to the form, focus is (h, k + a):
Final Answer: The focus is at (2, 0).
Try more like this with our Parabola Important Questions.
Practice Problems
- Find the focus of the parabola \( y^2 = 20x \).
- What is the focus of \( (y + 1)^2 = 8(x - 2) \)?
- Given \( x^2 - 6x + 8y = 0 \), calculate the vertex and focus.
- If the focus is at (0, 5) and axis is parallel to y-axis, write the equation of the parabola.
Common Mistakes to Avoid
- Switching the x and y forms—always check which variable is squared.
- Forgetting to adjust h and k if the parabola is shifted from the origin.
- Not dividing the coefficient by 4 to get the correct value of ‘a’.
Real-World Applications
The concept of focus of parabola is used in designing satellite dishes, car headlights, and in physics problems on reflection. See Conic Section Parabola for even more real-life links. Vedantu teaches students how understanding the focus connects maths to technology and design.
We explored the idea of focus of parabola, covered formulas, solved typical questions, and looked at its uses beyond exams. Practice more at Vedantu and check out Equation of Parabola or Analytic Geometry for further mastery.
FAQs on Understanding the Focus and Directrix of a Parabola
1. What is a focus and directrix of a parabola?
The focus of a parabola is a fixed point used in its geometric definition, while the directrix is a fixed straight line. A parabola consists of all points that are equidistant from the focus and the directrix. The vertex of the parabola lies midway between the focus and directrix.
2. How do you find the focus point of a parabola?
To find the focus of a parabola, first write the equation in standard form. For a parabola y2 = 4ax, the focus is at (a, 0). For x2 = 4ay, the focus is at (0, a). Rearranging the equation and identifying the values lets you locate the focus coordinates.
3. What is the value of 4p in a parabola?
In the standard equation of a parabola, such as y2 = 4ax or x2 = 4ay, the term 4p represents the distance between the vertex and the focus multiplied by four. Here, p is the distance from the vertex to the focus or the directrix.
4. What is the focal length of a parabola?
The focal length (usually denoted by p or a) of a parabola is the distance between the vertex and the focus. For a parabola y2 = 4ax, the focal length is 'a'.
5. How do you find the focus of a parabola given by the equation y2 = 4ax?
For the equation y2 = 4ax, the focus is located at (a, 0). Here, 'a' is one-fourth of the coefficient of x on the right side of the equation.
6. What is the directrix of a parabola and how is it calculated?
The directrix is a fixed line used in the definition of the parabola. For y2 = 4ax, the directrix is the line x = -a. For x2 = 4ay, the directrix is y = -a.
7. How do you find the vertex of a parabola?
The vertex is the turning point of the parabola and is found by converting the equation to standard form. For y2 = 4ax or x2 = 4ay, the vertex is at (0, 0). For other forms, complete the square to identify the vertex coordinates.
8. How do you find the focus of a parabola from its general equation?
To find the focus from a general parabola equation, first rewrite it in standard form. Identify the vertex, and use the value of 'a' or 'p' to locate the focus relative to the vertex along the axis of symmetry.
9. What are the focus and directrix of the parabola y2 = 8x?
For y2 = 8x, compare with y2 = 4ax to get a = 2. Therefore, the focus is at (2, 0), and the directrix is the line x = -2.
10. How do you find the focus of the parabola x2 = 4ay?
For the equation x2 = 4ay, the focus is at (0, a), and the directrix is y = -a.
11. What are the coordinates of the focus of y2 = 4x?
For y2 = 4x, a = 1. So, the focus is located at (1, 0).
12. How to find the focus of a parabola with the equation 4x2 - 12x + 8y + 13 = 0?
Rewrite as a standard form: 4x2 - 12x = -8y - 13, or x2 - 3x = -2y - 13/4. Complete the square: (x - 1.5)2 = -2y + 2.25 - 13/4, then rearrange to identify the vertex and value of a. Use these to find the focus coordinates. The process involves converting to standard form, identifying the vertex, and adding the focal length along the axis of symmetry from the vertex for the final focus position.
