Content Writer
Difference between Parabola and Hyperbola is based on the concept of eccentricity. Parabola and Hyperbola are a part of NCERT Class 11 Mathematics books.
- The difference between Parabola and Hyperbola is included in the topic of conic sections.
- When a plane intersects a cone at a specific angle, a curve is obtained, which is called a conic section.
- There are four types of conic sections: Parabola, Ellipse, Hyperbola, and Circle.
- A parabola is a form of plane curve that is approximately U-shaped.
- A hyperbola is a form of open curve where the plane is not parallel to the axis of the cone.
- Eccentricity is defined as the ratio between the centre to a vertex and from the centre to a focus.
- The distance between two fixed points is called foci.
- The general equation for parabola and Hyperbola are as follows:
y2 = 4ax
x2/a2 – y2/b2 = 1
- x is the transverse axis of hyperbola
- y is the conjugate axis of hyperbola
Key Terms: Difference between Parabola and Hyperbola, Parabola, Hyperbola, Eccentricity, Conic Sections, Foci, Ellipse, Circle, Vertex, Cone
What is a Parabola?
[Click Here for Sample Questions]
In a parabola, all points in a line are equidistant from a fixed line and a point. It is generated when a plane cuts through a conical surface parallel to the axis.
- Para means for, and bola means throwing.
- The eccentricity of the parabola is one.
- A parabola has one focus and one detrix.
- All figures are similar in shape but may differ in size.
- It will form a U-shaped curve.
- The vertex of a parabola is determined when the curve reaches the maximum or minimum point.
Formula for Parabola
The important formula for parabola are as follows:
| Form of the parabola | y2 = 4ax (Horizontal) | x2 = 4by (Vertical) |
| Vertex | (0,0) | (0,0) |
| Focus | (a,0) | (0,b) |
| Equation of the Directrix | x = -a | y = -b |
| Equation of the axis | y = 0 | x = 0 |
| Tangent at the vertex | x = 0 | y = 0 |
| Equation of latus rectum | x = a | y = b |
| Length of latus rectum | |4al | |4bl |
| End points of latus rectum | (a,2a) & (a,-2a) | (2b,b) & (-2b,b) |
Example of What is a Parabola?Example. The equation of parabola is y2 = 80x. Find the length of the latus rectum. Ans. Length of latus rectum, focus and vertex of the parabola is as follows: Given: Equation of a parabola: y2 = 80x Therefore, 4a = 80 a = 80/4 = 20 Now, parabola formula for latus rectum is: Length of latus rectum = 4a = 4(20) = 80 |
Parabola
Read More:
| Chapter Related Concepts | ||
|---|---|---|
| Angle Between Two Lines | Parabola Formula | Latus Rectum of Parabola |
What is a Hyperbola?
[Click Here for Sample Questions]
In a hyperbola all points in a line the distance from two fixed points is constant. An equilateral hyperbola is called where a=b. There are two focuses and two detrix in a hyperbola.
- The eccentricity of a hyperbola is more than one.
- Hyperbolas differ in size and shape.
- It is a form of curve that will create two mirror images of each other.
- The two mirror images are called connected components.
- In these curves, distance between two foci is equivalent to constant value.
- It can form many angles between the axis and the plane.
- Branches or curves are not parallel to each other.
Formula for Hyperbola
The important formula for hyperbola are as follows:
(x−x0)2/a2 – (y-y0)2/b2 = 1
- where, x0, y0 = the centre points
- a = semi-major axis
- b = semi-minor axis
The general equation used for hyperbola is as follows:
x2/a2 – y2/b2 = 1
- where, x is the transverse axis of hyperbola
- y is the conjugate axis of hyperbola
Example of What is a Hyperbola?Example. What will be the equation for the hyperbola which has center at (1, 9), vertex at (2, 11), and the focus at (5, 4). Ans. As we know, for hyperbola, all three points i.e., center, vertex, and focus lie on the same line y = 3. Now we can see from the given points: a = 1, c = 11 Hence b2 = c2- a2 = 11 – 1 = 10. Putting in the equation of hyperbola conic section: (x−h)2/a2 - (y−k)2/b2 = 1 We get, (x−1)2/12 - (y−3)2/10 = 1 |
Hyperbola
Difference between Parabola and Hyperbola
[Click Here for Sample Questions]
The difference between parabola and hyperbola are as follows:
| Parabola | Hyperbola |
|---|---|
| In a Parabola, all points in a line are equidistant from a fixed line and a point. | In a Hyperbola all points in a line the distance from two fixed points is constant. |
| A parabola has one focus and one detrix. | There are two focuses and two detrix in a hyperbola. |
| The eccentricity of the parabola is one. | The eccentricity of a hyperbola is more than one. |
| All parabolas are similar in shape but may differ in size. | Hyperbolas differ in size and shape. |
| The arms of the parabola are parallel to each other. | The arms of the hyperbola are not parallel to each other |
| Parabola has zero asymptotes | Hyperbola has two asymptotes |
| The standard equation of a parabola is y2 = 4ax | The standard equation of hyperbola is: x2-a2/y2-b2 = 1 |
Things to Remember
- Difference between Parabola and Hyperbola determine difference between a set of points.
- A parabola has one focus and one detrix.
- On the other hand there are two focuses and two detrix in a hyperbola.
- The eccentricity of the parabola is one and that of a hyperbola is more than one.
- Since exams are approaching, candidates can solve NCERT Solutions For Class 11 Maths Chapter 11: Conic Sections.
Read More:
| Class 11 Mathematic Related Concept | ||
|---|---|---|
| Value of e | Maxima and Minima | Eccentricity of an Ellipse |
Sample Questions
Ques: Define the four types of conic sections? (2 marks)
Ans: There are four types of conic sections: Parabola, Ellipse, Hyperbola, and Circle. In a Parabola, all points in a line are equidistant from a fixed line and a point. In a Hyperbola all points in a line the distance from two fixed points is constant. In Ellipse the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
Ques: What is eccentricity? (2 marks)
Ans: Eccentricity is the distance from one point to the focus which is divided by the distance which is perpendicular to the nearest detrix. The eccentricity of the parabola is one and the eccentricity of a hyperbola is more than one.
Ques: What is the major difference between parabola and hyperbola? (2 marks)
Ans: The major difference between parabola and hyperbola is because of the eccentricity of the curves. : Eccentricity is the distance from one point to the focus which is divided by the distance which is perpendicular to the nearest detrix. The eccentricity of the parabola is one and the eccentricity of a hyperbola is more than one.
Ques: What is the equation of the parabola with focus (4,0) and detrix x is -4? (2 marks)
Ans: The focus (4, 0) lies on the x-axis which lies on the parabola. The equation of the parabola is y2 = 4ax. The required equation is y2= 16x.
Ques: What is the equation of the parabola which is symmetric above the y axis and passes through the point(2,-3)? (3 marks)
Ans: The parabola is symmetric along the y-axis so the equation of a parabola is x2 = 4ay or x2= -4ay. Since the parabola passes through the point (2, -3) the equation of a parabola is x2 = -4ay.
Since the parabola passes through the point (2, -3):
4 = -4a (-3)
a = 1/3 .
Thus the equation of parabola is 3x2 = -4y.
Ques: The equation of the hyperbola is given as (x - 3)2/62 - (y - 2)2/ 42 = 1. Find the asymptote of this hyperbola? (3 Marks)
Ans: The formula for the asymptotes of a hyperbola is:
y = y0 + b/ax – b/ax0
y = y0 − b/ax + b/ax0
Using the above formula
y = 2 - (4/6)x + (4/6)5 and y = 2 + (4/6)x - (4/6)5
Asymptotes are y = 2 - (4/6)x + 4, and y = 2 + (4/6)x - 4.
Ques: Find the equation of the parabola whose focus is (2,-4) and directrix, x - y + 5 = 0? (3 Marks)
Ans: Let as assume P(x, y) as any point on the parabola. Then
=> √(x-2)2 + (y+4)2 = |x-y+5|/√1+1
=> (x-2)2 + (y+4)2 = (x-y+5)2/2
=> x2 + y2 + 2xy - 22x + 26y + 25= 0
=> So, (x+y)2 = 22x - 26y - 25
Ques: Find the length of the latus rectum of the parabola y2 = 4ax if the line x + 3y = 1 touches it? (3 Marks)
Ans: Tangent equation to y2 = 4ax is written as :
=> y = mx + a/m
=> m2x - my+ a =0
Comparing this equation with the given tangent (2x + 3y - 1 = 0), we find
m2/1 = -m/3 = a/-1
=> m= -? , a = m/3 = -1/9
Hence, the length of the latus-rectum
= 4*a = 4/9, (Ignore the negative sign)
Ques: The equation of parabola is y2 = 160x. Find the length of the latus rectum, focus, and vertex? (3 marks)
Ans: Length of latus rectum, focus and vertex of the parabola is as follows:
Given: Equation of a parabola: y2 = 40x
Therefore, 4a = 160
a = 160/4 = 40
Now, parabola formula for latus rectum is:
Length of latus rectum = 4a
= 4(40) = 160
Now, focus= (a,0) = (40,0)
Now, Vertex = (0,0)
Ques: The equation of the hyperbola is given as (x - 5)2/92 - (y - 2)2/ 52 = 1. Find the length of the Major Axis and Minor Axis? (3 marks)
Ans: According to the formulas:
- Length of the major axis = 2a, and the length of the minor axis = 2b
- Length of the major axis = 2 × 9 = 18, and the length of the minor axis = 2 × 5 = 10
Ques: Find the length of the latus rectum of the parabola y2 = 4ax if the line 5x + 3y = 1 touches it? (3 Marks)
Ans: Tangent equation to y2 = 4ax is written as :
=> y = mx + a/m
=> m2x - my+ a =0
Comparing this equation with the given tangent (2x + 3y - 1 = 0), we find
m2/5 = -m/3 = a/-1
=> m= -? , a = m/3 = -5/9
Hence, the length of the latus-rectum
= 4*a = 20/9, (Ignore the negative sign)
Read Also:
Comments