Parabola definition standard equation vertex focus directrix and solved examples
In mathematics, the intersection of the surface of a cone with a plane is called Conic Section or simply Conic.
There are three types of conic section; the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type.
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In this article we will all discuss the Basics of Parabola, so let's begin.
Parabola !!
When we cut cones in the specific manner as shown in above figure, then we get the shape of Parabola.
Mathematical Definition : A parabola is a locus of points which are at an equal distance from: a fixed point and a fixed straight line. Fixed point is called Focus of parabola and the Fixed line is called Directrix.
Terms Related to Parabola
Directrix - It is the line perpendicular to the axis of symmetry but not passes through the focus, the distance of any point on the curve from the directrix will be equal to the distance from the focus of the same point.
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Focus - It is a special point which lies on the axis of symmetry and which defines the parabola. We also say that Parabola is a special case of ellipse with one of the foci at the point of infinity.
Axis of Symmetry - As per the name It is a line of symmetry which divides the parabola in two equal parts. It is also a perpendicular to the directrix which passes through the focus.
Vertex - At vertex the parabola will be abruptly curved. Vertex is the point where the parabola and the axis of symmetry intersects with each other.
Focal Length - It is the distance between the focus on the axis of symmetry and the vertex.
Latus Rectum - It is the line segment which is parallel to the directrix and passes through the focus of the parabola.
Equations of Parabola
The general standard form of the equation of parabola is given by
$y^2 = 4ax$
where a is the distance of the focus from the origin (and also of the directrix from the origin). In the below graph you will be able to visualize these things.
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There are 4 possibilities of the curve on the basis of quadrant and axis of symmetry, these are described below.
$Y^2 = 4aX $, (where $X,Y$ are variables and $a$ is constant.)
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$Y^2 = - 4aX $
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$X^2 = 4aY $
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$X^2 = - 4aY$
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Properties of Parabola :
Focus of Parabola: Focus is a special point from which the distance is measured to form conic and this defines the parabola. The parabola has the vertex as the midpoint of the line segment between focus and the directrix.
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Latus Rectum of Parabola: It is the line which is parallel to the directrix and which passes through the focus of the parabola. It will be a perpendicular to the axis of symmetry. In parabola, let $a$ is the distance of the vertex from the focus, which is also at equal distance from the vertex to directrix, so the distance from focus to directrix will be $2a$. The distance from the focus to the point on the plane will be equal from focus to directrix that is, $2a$. Focus lies on the axis of symmetry which is also the midpoint of the latus rectum, and which shows that the half of the latus rectum is $2a$ and the length of latus rectum is $4a$.
Eccentricity of Parabola: Eccentricity of conic sections is the factor which describes how much circular the conic section is. More eccentricity will show less spherical and less eccentricity will show more spherical. It will be denoted by $”e”$. More mathematically, The eccentricity of parabola is the ratio of the distance between the focus and a point on the plane to the directrix and that same point only. As the parabola is a locus of all the points which are at equal distance from the directrix and the focus, this ration will always be $1$ that means, $e = 1$.
Important Table to Remember
Parabola: The locus of a point which moves such that its distance from a fixed point is equal to its distance from a fixed straight line, i.e., $e = 1$ is called a parabola.
Applications of Conic Sections
To predict the position and path of the paths of the planets around the sun
Parabolic mirrors are to be used to concentrate the light beams at the focus of the parabola
Parabolic microphones perform a similar concept with the sound waves
Solar ovens use parabolic mirrors to concentrate the light beams to use for heating
In the design of car headlights and in spotlights, the parabola is used because it aids in converging the light beam
the trajectory of objects which is thrown or shot near to the earth's surface is predicted by its parabolic path
Hyperbolas are generally used in the navigation system which is known as LORAN (long range navigation)
Hyperbolic and Parabolic mirrors and lenses are generally used in various systems of telescopes
FAQs on Conic Section Parabola Explained with Equations and Graphs
1. What is a parabola in conic sections?
A parabola is the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. In conic sections, a parabola is formed when a plane cuts a right circular cone parallel to its slant height. Key elements of a parabola include:
- Vertex – the turning point of the curve
- Focus – fixed point inside the curve
- Directrix – fixed line outside the curve
- Axis of symmetry – line passing through vertex and focus
2. What is the standard equation of a parabola?
The standard equation of a parabola depends on its orientation and vertex position. For a vertex at the origin:
- Right-opening: y² = 4ax
- Left-opening: y² = −4ax
- Upward-opening: x² = 4ay
- Downward-opening: x² = −4ay
3. What is the focus and directrix of a parabola?
The focus is a fixed point inside the parabola and the directrix is a fixed line used to define the curve. For example, in y² = 4ax:
- Focus = (a, 0)
- Directrix = x = −a
4. How do you find the vertex of a parabola?
The vertex of a parabola is its turning point and can be found directly from its equation. For:
- Standard form like y² = 4ax, the vertex is (0, 0).
- General form y = ax² + bx + c, vertex x-coordinate is −b/2a.
- x = −(−4)/(2×2) = 1
- Substitute x = 1 → y = −1
- Vertex = (1, −1)
5. What is the difference between a parabola and an ellipse?
The main difference is that a parabola has one focus while an ellipse has two foci. Key differences include:
- Parabola eccentricity: e = 1
- Ellipse eccentricity: 0 < e < 1
- Parabola is open curve
- Ellipse is closed curve
6. How do you know if a quadratic equation represents a parabola?
A quadratic equation in one variable, such as y = ax² + bx + c (where a ≠ 0), always represents a parabola. In coordinate geometry:
- If only one variable is squared (x² or y²), it represents a parabola.
- If both variables are squared with same sign, it represents a circle or ellipse.
7. What is the latus rectum of a parabola?
The latus rectum of a parabola is the line segment passing through the focus and perpendicular to the axis of symmetry. For the parabola y² = 4ax:
- Length of latus rectum = 4a
- Endpoints are (a, 2a) and (a, −2a)
8. How do you find the equation of a parabola given the focus and directrix?
To find the equation of a parabola, use the definition that distance from focus equals distance from directrix. Steps:
- Let (x, y) be any point on the parabola.
- Set distance to focus = distance to directrix.
- Simplify to standard form.
- Distance formula gives y² = 8x
- Final equation: y² = 8x
9. What are the key properties of a parabola?
The key properties of a parabola in conic sections describe its symmetry and geometric behavior. Important properties include:
- It is symmetric about its axis of symmetry.
- Distance from focus equals distance from directrix.
- Eccentricity e = 1.
- It reflects parallel rays to the focus (reflective property).
10. What are real-life applications of a parabola?
A parabola is widely used in real life due to its reflective property and quadratic shape. Common applications include:
- Satellite dishes and antennas
- Car headlights and torch reflectors
- Projectile motion in physics
- Suspension bridges cables
