What is the Conic Section? Define!
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
1. What is a parabola as defined in the chapter on Conic Sections?
A parabola is a type of conic section, formed by the intersection of a cone with a plane that is parallel to the cone's side. In geometric terms, a parabola is the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed straight line, called the directrix. The eccentricity of any parabola is exactly 1.
2. What are the standard equations of a parabola for Class 11?
For a parabola with its vertex at the origin (0,0), there are four standard equations based on its orientation:
- y² = 4ax: This parabola opens to the right, with the x-axis as its axis of symmetry.
- y² = -4ax: This parabola opens to the left, also with the x-axis as its axis of symmetry.
- x² = 4ay: This parabola opens upwards, with the y-axis as its axis of symmetry.
- x² = -4ay: This parabola opens downwards, also with the y-axis as its axis of symmetry.
In these equations, 'a' is a positive constant representing the distance from the vertex to the focus.
3. What are the main components used to describe a parabola?
The key components that define a parabola's geometry and position are:
- Vertex: The point on the parabola where it intersects its axis of symmetry. It is the point where the curve makes its sharpest turn.
- Focus: A fixed point inside the parabola that is used to define the curve's shape.
- Directrix: A fixed line outside the parabola. Every point on the parabola is an equal distance from the focus and the directrix.
- Axis of Symmetry: The line that passes through the vertex and focus, dividing the parabola into two mirror-image halves.
- Latus Rectum: The chord passing through the focus and perpendicular to the axis of symmetry.
4. What is the latus rectum of a parabola, and what does its length signify?
The latus rectum of a parabola is the line segment that passes through the focus, is perpendicular to the axis of symmetry, and has both its endpoints on the parabola. Its length is a measure of the parabola's 'width' at the focus. For any standard parabola (e.g., y² = 4ax or x² = 4ay), the length of the latus rectum is constant and is equal to 4a.
5. What are some common real-life examples of parabolas?
Parabolic shapes are common in science and engineering due to their unique reflective properties. Key examples include:
- Satellite Dishes: A parabolic dish collects parallel radio waves and reflects them to a single point, the focus, where the receiver is placed.
- Car Headlights: A bulb placed at the focus of a parabolic mirror reflects light into a strong, parallel beam.
- Projectile Motion: The path an object follows when thrown into the air (e.g., a ball) is a parabola, assuming air resistance is negligible.
- Suspension Bridges: The main cables of many suspension bridges hang in a shape that is very close to a parabola.
6. How does the value 'a' in the equation y² = 4ax influence the shape of the parabola?
The parameter 'a' in the equation y² = 4ax directly controls the 'width' of the parabola. Since 'a' is the distance from the vertex to the focus, a larger value of 'a' places the focus further from the vertex. This results in a wider, more open parabola. Conversely, a smaller value of 'a' brings the focus closer to the vertex, creating a narrower parabola.
7. Why is a parabola defined using a focus and a directrix?
Defining a parabola by the equidistant relationship between its focus and directrix is crucial because this geometric property is responsible for its unique reflective capabilities. This precise definition ensures that any ray travelling parallel to the axis of symmetry will reflect off the parabola and pass through the focus. This property is fundamental to the design of antennas, telescopes, and reflectors.
8. What is the primary difference between a parabola like y² = 4ax and one like x² = 4ay?
The main difference between these two standard forms is their orientation and axis of symmetry. A parabola of the form y² = 4ax is symmetric about the x-axis and opens horizontally (sideways). In contrast, a parabola of the form x² = 4ay is symmetric about the y-axis and opens vertically (upwards or downwards). The roles of x and y are effectively swapped, determining the direction of the opening.
9. How does a parabola fit in with other conic sections like circles and ellipses?
A parabola is one of the four types of conic sections, all of which are defined by a property called eccentricity (e). Eccentricity measures how much a curve deviates from being circular. The relationship is as follows:
- An ellipse has an eccentricity between 0 and 1 (0 < e < 1).
- A circle is a special ellipse with an eccentricity of exactly 0 (e = 0).
- A parabola has an eccentricity of exactly 1 (e = 1).
- A hyperbola has an eccentricity greater than 1 (e > 1).
Therefore, a parabola can be thought of as the boundary case between the closed curve of an ellipse and the open curves of a hyperbola.
