How to Derive and Use the Standard Equation of a Parabola?
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 with Graphs
1. What is the standard equation of a parabola?
The standard equation of a parabola algebraically represents the parabola with its vertex and orientation. When the vertex is at the origin, typical forms are y² = 4ax (parabola opening right), y² = -4ax (opening left), x² = 4ay (opening upwards), and x² = -4ay (opening downwards). These equations help in plotting, solving, and understanding the parabola's geometric properties.
2. How do you derive the standard parabola equation?
The derivation of the standard equation of a parabola starts from the geometric definition: a parabola is the set of points equidistant from the focus and the directrix. By using the distance formula and setting the distance from any point (x,y) to the focus equal to the distance to the directrix, we arrive at the equation y² = 4ax or x² = 4ay for parabolas with vertices at the origin. This stepwise approach clarifies the roles of focus, directrix, and parameter a.
3. What does 4a represent in the equation?
In the standard parabola equation, 4a represents the distance related to the parabola's shape. Specifically, a is the distance from the vertex to the focus, and 4a is the length of the latus rectum, the chord passing through the focus and perpendicular to the axis of symmetry. This parameter controls how wide or narrow the parabola opens.
4. How does the vertex position affect the equation?
When the vertex is not at the origin but at point (h, k), the standard equation transforms to (y - k)² = 4a(x - h) or (x - h)² = 4a(y - k), depending on the parabola's orientation. This vertex shift moves the parabola along the coordinate plane without changing its shape, making it essential for solving problems with translated parabolas.
5. What is the difference between general and standard forms?
The standard form of a parabola explicitly shows the position of the vertex and focus with simplified orientation along axes, such as y² = 4ax. The general form is a quadratic equation like Ax² + Bxy + Cy² + Dx + Ey + F = 0 that can represent any conic section, including parabolas, ellipses, and hyperbolas. Converting the general to standard form involves completing the square and rotating axes if needed.
6. What are the main types of parabolas and their equations?
There are four principal types of parabolas based on their orientation:
• Parabola opening right: y² = 4ax
• Parabola opening left: y² = -4ax
• Parabola opening upwards: x² = 4ay
• Parabola opening downwards: x² = -4ay
Each type has a specific standard equation reflecting its axis of symmetry and direction.
7. Why can the standard equation have positive or negative 4a?
The sign of 4a in the standard equation indicates the direction the parabola opens relative to the coordinate axes. A positive 4a means it opens towards the positive x-axis or y-axis, while a negative 4a means it opens in the opposite direction. This sign distinction helps in correctly graphing and solving parabolic equations.
8. Why do some students confuse focus with vertex in equations?
Confusion arises because the vertex and focus are closely related but distinct points. The vertex is the parabola's turning point, while the focus lies at a distance a from the vertex along the axis of symmetry. Students may mistake the vertex coordinates for the focus or vice versa, leading to errors in plotting or formulating the equation. Emphasizing their definitions and roles clears this confusion.
9. Can a parabola open in any direction not aligned with the x or y axis?
In standard coordinate geometry, parabolas are typically aligned along the x or y axes. However, parabolas can open in any direction if the coordinate axes are rotated, requiring the general form with an xy term. These rotated parabolas need coordinate transformation techniques, which are beyond basic standard equations but important for advanced studies.
10. What mistakes happen in shifting the vertex from (0,0) to (h,k)?
Common mistakes when shifting the vertex include:
• Incorrect substitution of (x - h) and (y - k) in the equation.
• Forgetting to change both variables accordingly.
• Misidentifying the new focus and directrix positions.
Proper understanding of translation and maintaining the parabola’s orientation is essential for accuracy.
11. Why is the latus rectum length important in the equation?
The latus rectum is a chord through the focus, perpendicular to the axis of symmetry, with length 4a. It serves as a key parameter indicating the width of the parabola near the focus. Knowing the latus rectum length helps in graphing the parabola precisely and solving related geometric problems.
12. How is the focus and directrix related in the standard parabola equation?
The focus and directrix are fundamental to defining a parabola. The standard equation ensures each point on the parabola has equal distance to the focus and the directrix line. For example, in y² = 4ax, the focus is at (a, 0) and the directrix is the vertical line x = -a. This relationship mathematically encodes the geometric property of the parabola.
