What Is the Focus of a Parabola Formula Derivation and Solved Examples
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 Focus of Parabola Explained with Definition and Formula
1. What is the focus of a parabola?
The focus of a parabola is a fixed point inside the curve such that every point on the parabola is equidistant from the focus and the directrix. In simple terms:
- A parabola is the set of all points whose distance from the focus equals their distance from a fixed line called the directrix.
- The focus lies on the axis of symmetry of the parabola.
- It determines the shape and opening of the parabola.
2. What is the formula for the focus of a parabola?
The focus formula depends on the standard form of the parabola’s equation.
- For y² = 4ax, focus = (a, 0)
- For x² = 4ay, focus = (0, a)
- For y² = -4ax, focus = (-a, 0)
- For x² = -4ay, focus = (0, -a)
Here, a is the distance from the vertex to the focus.
3. How do you find the focus of a parabola from its equation?
To find the focus of a parabola, first convert the equation into standard form and then identify the value of a.
- Step 1: Rewrite the equation in the form y² = 4ax or x² = 4ay.
- Step 2: Compare and find a.
- Step 3: Use the focus formula.
Example: For y² = 8x → 4a = 8 → a = 2 → Focus = (2, 0).
4. Where is the focus located in a parabola?
The focus is located on the axis of symmetry at a distance a from the vertex inside the parabola.
- If the parabola opens right → focus is to the right of the vertex.
- If it opens left → focus is to the left.
- If it opens upward → focus is above the vertex.
- If it opens downward → focus is below the vertex.
The exact position depends on the sign in the standard equation.
5. What is the distance between the vertex and the focus?
The distance between the vertex and the focus is equal to a in the standard equation.
- In y² = 4ax or x² = 4ay, the distance = |a|.
- If 4a = 12, then a = 3, so the distance is 3 units.
This distance controls how wide or narrow the parabola appears.
6. What is the relationship between the focus and directrix of a parabola?
A parabola is defined as the set of points that are equidistant from the focus and the directrix.
- The focus is a fixed point.
- The directrix is a fixed straight line.
- Distance from any point on the parabola to the focus equals its perpendicular distance to the directrix.
This geometric property defines the shape of the parabola.
7. How do you find the focus of a parabola in vertex form?
To find the focus in vertex form, use the formulas derived from (y − k)² = 4a(x − h) or (x − h)² = 4a(y − k).
- For (y − k)² = 4a(x − h), focus = (h + a, k)
- For (x − h)² = 4a(y − k), focus = (h, k + a)
Example: (y − 1)² = 8(x − 2) → 4a = 8 → a = 2 → Focus = (4, 1).
8. Can the focus of a parabola be outside the curve?
No, the focus of a parabola always lies inside the curve along its axis of symmetry.
- The parabola opens away from the directrix.
- The focus lies on the inner side of the curve.
- Its position determines the reflective property of the parabola.
It cannot lie outside the parabola’s open region.
9. What is the focus of the parabola x² = 16y?
The focus of x² = 16y is (0, 4).
- Compare with standard form x² = 4ay.
- 4a = 16 → a = 4.
- Focus = (0, a) = (0, 4).
The parabola opens upward since the coefficient is positive.
10. Why is the focus important in real life applications of a parabola?
The focus is important because parabolas reflect all incoming rays parallel to the axis through the focus.
- Used in satellite dishes to collect signals at the focus.
- Used in car headlights to project parallel beams of light.
- Used in solar cookers to concentrate heat at the focus.
This reflective property makes the focus crucial in engineering and physics applications.
