What Are the Standard Equations of Parabola with Vertex Focus and Examples
The concept of Standard Equations of Parabola is essential in mathematics and helps in solving problems related to conic sections, coordinate geometry, and competitive exams like JEE and board tests.
Understanding Standard Equations of Parabola
A standard equation of a parabola represents the algebraic form of a parabola when its vertex and axis are positioned at convenient locations, usually at the origin and along the axes. This concept is widely used in coordinate geometry, quadratic equations, and the study of conic sections. Knowing the standard equation of parabola helps to identify the vertex, focus, directrix, and the orientation (whether it opens right, left, up, or down) easily.
Standard Equations of Parabola – Formulae
The standard equation depends on the orientation and position of the parabola's vertex. The primary standard forms are:
| Position/Orientation | Standard Equation | Opens |
|---|---|---|
| Vertex at (0,0), axis along x-axis | \( y^2 = 4ax \) | Right |
| Vertex at (0,0), axis along x-axis | \( y^2 = -4ax \) | Left |
| Vertex at (0,0), axis along y-axis | \( x^2 = 4ay \) | Upward |
| Vertex at (0,0), axis along y-axis | \( x^2 = -4ay \) | Downward |
| Vertex at (h,k), axis parallel to x-axis | \( (y-k)^2 = 4a(x-h) \) | Right/Left (depends on sign) |
| Vertex at (h,k), axis parallel to y-axis | \( (x-h)^2 = 4a(y-k) \) | Up/Down (depends on sign) |
These equations are termed as standard equations of parabola and are most useful when the vertex is at the origin or at a point (h, k) in the coordinate plane.
Derivation of Standard Equation of Parabola
To derive the standard equation, let's consider a parabola that opens to the right, with vertex at the origin (0,0), focus at (a, 0), and directrix x = -a:
1. By definition, any point (x, y) on the parabola is equally distant from the focus (a, 0) and the directrix x = -a.
2. Distance from (x, y) to focus (a, 0): \( \sqrt{(x-a)^2 + y^2} \)
3. Distance from (x, y) to the directrix x = -a: \( |x + a| \)
4. Set these distances equal: \( \sqrt{(x - a)^2 + y^2} = |x + a| \)
5. Squaring both sides: \( (x - a)^2 + y^2 = (x + a)^2 \)
6. Expanding: \( x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2 \)
7. Subtract \( x^2 + a^2 \) from both sides: \( -2ax + y^2 = 2ax \)
8. Bringing like terms together: \( y^2 = 4ax \)
This is the standard equation of a parabola opening to the right.
Key Elements in the Standard Equation
- Vertex: The fixed point (h, k) about which the parabola is symmetrical.
- Axis: The line passing through the vertex; parabola is symmetric about this axis.
- Focus: Point from which distances are measured (e.g., (a, 0) in \( y^2 = 4ax \)).
- Directrix: Line (e.g., x = -a) used for geometric definition.
Worked Example – Solving Standard Equation Problems
Let's solve an example for the standard equation of parabola:
1. Given: Parabola \( y^2 = 12x \).
2. Compare with standard form \( y^2 = 4ax \):
- \( 4a = 12 \implies a = 3 \)
3. The parabola opens to the right (since coefficient is positive).
4. Focus: (a, 0) = (3, 0)
5. Directrix: x = –a = –3
6. Length of latus rectum: \( 4a = 12 \)
Final answers: Focus: (3, 0); Directrix: x = –3; Latus rectum: 12
Practice Problems
1. Find the standard equation of the parabola with focus at (4, 0) and directrix x = –4.
2. A parabola has its vertex at (0, 0) and passes through (5, 2), symmetric about the y-axis. Write its standard equation.
3. Given (x – 2)^2 = –8(y – 3), state the direction it opens and find vertex, focus, directrix.
Common Mistakes to Avoid
- Mixing up the x and y in the standard equations for different orientations.
- Forgetting to check the sign of 4a to decide the opening direction.
- Not shifting correctly when the vertex is not at (0,0).
- Confusing the value of 'a' with latus rectum or focus distance.
Real-World Applications
The standard equation of parabola is extremely useful in physics (projectile motion), engineering (reflectors, antennas), and computer graphics. Students applying this concept can solve practical geometry problems required in board and competitive exams. Vedantu’s maths resources make it easier to visualize and excel in these topics.
Summary Table: All Standard Forms with Elements
| Equation Form | Vertex | Focus | Directrix | Latus Rectum |
|---|---|---|---|---|
| \( y^2 = 4ax \) | (0,0) | (a,0) | x = –a | 4a |
| \( (y-k)^2 = 4a(x-h) \) | (h,k) | (h+a, k) | x = h–a | 4a |
| \( x^2 = 4ay \) | (0,0) | (0,a) | y = –a | 4a |
| \( (x-h)^2 = 4a(y-k) \) | (h,k) | (h, k+a) | y = k–a | 4a |
Review this table for competitive/revision preparation needs. For more on related graphing concepts, check out the Parabola Graph resource or compare equations at Equation of Parabola on Vedantu.
We explored the idea of standard equations of parabola, how to apply each formula, common pitfalls, and solved examples. Practice regularly and refer to Vedantu for personalized maths guidance!
Related Topics: Properties of Parabola, Latus Rectum, Conic Section Parabola
FAQs on Standard Equations of a Parabola Explained Clearly
1. What is the standard equation of a parabola?
The standard equation of a parabola depends on its orientation and is written as y² = 4ax or x² = 4ay. These forms represent parabolas whose vertex is at the origin.
- y² = 4ax → opens right (if a > 0) or left (if a < 0)
- x² = 4ay → opens upward (if a > 0) or downward (if a < 0)
- a is the distance from the vertex to the focus
2. What is the formula of a parabola with vertex at the origin?
The formula of a parabola with vertex at the origin is y² = 4ax (horizontal axis) or x² = 4ay (vertical axis). These equations describe parabolas centered at (0,0).
- If axis is along x-axis → y² = 4ax
- If axis is along y-axis → x² = 4ay
- The vertex is always at (0,0)
3. What does the value of a represent in the standard equation y² = 4ax?
In the equation y² = 4ax, the value a represents the distance between the vertex and the focus. It determines the shape and direction of the parabola.
- Focus = (a, 0)
- Directrix = x = −a
- Larger |a| → wider parabola
- Smaller |a| → narrower parabola
4. How do you find the focus and directrix from the standard equation of a parabola?
The focus and directrix are found directly from the value of a in the standard equations y² = 4ax or x² = 4ay.
- For y² = 4ax:
- Focus = (a, 0)
- Directrix = x = −a
- For x² = 4ay:
- Focus = (0, a)
- Directrix = y = −a
5. What is the equation of a parabola that opens upward?
A parabola that opens upward has the standard equation x² = 4ay where a > 0. This means the axis of symmetry is vertical.
- Vertex = (0,0)
- Focus = (0,a)
- Directrix = y = −a
6. What is the difference between y² = 4ax and x² = 4ay?
The difference between y² = 4ax and x² = 4ay is the direction in which the parabola opens.
- y² = 4ax → opens left or right (horizontal axis)
- x² = 4ay → opens up or down (vertical axis)
- The squared variable shows the axis direction
7. How do you write the standard equation of a shifted parabola?
The standard equation of a shifted parabola is written as (y − k)² = 4a(x − h) or (x − h)² = 4a(y − k). Here, (h, k) is the vertex.
- Vertex = (h, k)
- Horizontal axis → (y − k)² = 4a(x − h)
- Vertical axis → (x − h)² = 4a(y − k)
8. Can you give an example of finding a from the equation of a parabola?
Yes, you can find a by comparing the given equation with y² = 4ax or x² = 4ay.
- Example: y² = 12x
- Compare with y² = 4ax
- 4a = 12
- a = 3
9. What is the length of the latus rectum of a parabola?
The length of the latus rectum of a parabola is 4a. It is the chord passing through the focus and perpendicular to the axis of symmetry.
- For y² = 4ax → length = 4a
- For x² = 4ay → length = 4a
10. How do you identify the direction of opening of a parabola from its equation?
The direction of opening is identified by checking which variable is squared and the sign of a in the standard equation.
- If equation is y² = 4ax:
- a > 0 → opens right
- a < 0 → opens left
- If equation is x² = 4ay:
- a > 0 → opens upward
- a < 0 → opens downward
