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A conic sections is a curve formed by intersecting a plane with a cone, known as the cutting plane. It is formed results when a cone is intersected by a plane. Cones are right circular when the axis passes through the base’s centre.
- Cones are formed at right angles to the plane.
- Oblique cones have an axis that does not travel perpendicularly through the centre of the base.
- We owe this knowledge about conic sections to Apollonius (200 BC).
- He wrote a book called The Conic, which led to many scientific and mathematical discoveries.
- It became an advancement in the history of sciences when it was used in Kepler’s work.
- Conic sections are also known as quadratic curves.
- Hyperbola, parabola and ellipse are three categories of the quadratic curve.
- The theory has been the main source of knowledge behind Euclidean geometry.
- To understand the conic sections, consider the case of planets revolving around the sun in elliptical order.
Read More: Parabola Graph: Standard Form of Parabola Equation
Key Terms: Conic Sections, Parabola, Hyperbola, Ellipse, Circle, Cone, Quadratic Curve, Euclidean Geometry, Locus, Focus, Eccentricity, Directrix, Vertex, Focal Chord
What are Conic Sections?
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Conic Sections are a form of curve that is formed by the intersection of a plane and a right circular cone. The intersection of the plane can be a parabola, hyperbola, ellipse or circle, depending upon the angle of the plane.
- Conic sections are also known as sections of a cone.
- When a plane is passing through an apex, it is considered to be the special case of the section.
- The vertex of the cone is divided into two nappes, namely upper nappes and lower nappes.
- It can form various shapes depending upon the cut between the plane and the cone and its nappe.
- The concept is widely used in the fields of physics, optical mechanics, and orbits.
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Example of What are Conic Sections?Example: Headlights used in cars are real life examples of conic sections. In this, the device is formed by a paraboloid of revolution about its axis.
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Read More: NCERT Solutions For Class 11 Maths Chapter 11: Conic Sections
Parameters of Conic Section
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Some important parameters used in the conic section are as follows:
Latus Rectum
Latus Rectum is a chord that runs parallel to the directrix and passes through a focus. The latus rectum of hyperbola and ellipse is 2b2/a and for parabola it is 4a.
Axis
Any line which passes through the focus and is perpendicular to the directrix is the axis.
Principal axis
The line connecting the two focal points or foci of an ellipse or hyperbola is known as the principal axis.
Major axis
Major Axis is a chord that connects the two vertices. It is an ellipse's longest chord.
Read More: Difference between Parabola and Hyperbola
Minor axis
The ellipse's shortest chord is called the minor axis.
Linear Eccentricity
Linear Eccentricity is the distance between a section's focus and its centre.
Focal Parameter
The distance between the focus and the accompanying directrix is known as the focal parameter.
Read More: Coplanarity two lines
Focus, Eccentricity and Directrix of Conic
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Focus, Eccentricity and Directrix are three important terms that are used in solving problems related to conic section. Even the shape and orientation of the figures are based on these three terms.
- A detailed discussion about these terms are provided in the below section:
Directrix
Directrix is a form of curve that is used for creating various geometric objects. With reference to the conic section, it is a line drawn perpendicular to the section.
- It is parallel to the conjugate axis.
- Each point on the conic section is the ratio of the distance of the directrix and the foci.
Directrix of various shapes in conic sections are as follows:
| Shape | Directrix |
|---|---|
| Circle | 0 |
| Ellipse | 2 |
| Parabola | 1 |
| Hyperbola | 2 |
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Eccentricity
It is a measure of how far the ellipse deviates from being circular. The eccentricity is the angle formed between the cone's surface and its axis or between the cutting plane and the axis:
e = cos α/cos β
- Eccentricity is defined as a non-negative number that describes the shape of the object.
Eccentricity of various shapes in conic sections are as follows:
| Shape | Eccentricity |
|---|---|
| Circle | 0 |
| Ellipse | Between 0 – 1 |
| Parabola | 1 |
| Hyperbola | Greater than 1 |
| Pair of lines | Infinity |
Focus
Focus are defined as special points with respect to which various shapes or curves are formed in a plane. It is also known as foci. A single focus can be used to describe entire conic sections.
Focus of various shapes in conic sections are as follows:
| Shape | Focus |
|---|---|
| Circle | Constant |
| Ellipse | 2 |
| Parabola | 1 |
| Hyperbola | 2 |
| Pair of lines | 2 |
Read More: Geometry Formula
Sections of a Cone
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There are four sections of a cone:
Circle
Circle is the locus of a point that is moving in a certain plane around a fixed distance. The equation of a circle with centre (h,k) and the radius r is:
(x–h)2 + (y–k)2 =r2
- The equation of a circle with radius r having centre (h, k) is given as
(x – h)2 + (y – k)2 = r2
- The general equation of the circle, where, g, f and c are constants is given as
x2 + y2 + 2gx + 2fy + c = 0
- The general equation of the circle passing through origin.
x2 + y2 + 2gx + 2fy = 0
- The radius of the circle, r, where, (-g, -f) is the centre of the circle.
\(\sqrt{g^2+f^2-c}\)
Read More: Properties of Parallel Lines
Ellipse
Ellipse is the set of all points in a plane, the sum of whose distance from two fixed points in a plane is constant. With the foci on the x-axis, the equation of an ellipse can be written as:
Example of EllipseExample. If for an ellipse, the focus lies at (10, 0), a vertex lies at (8, 0), and its center lies at (0, 0). Find the equation of the ellipse. Ans. From the given points, we can see that c = 10 and a = 8. Using b2 = a2 – c2 We get: b2= 100 – 64 = 36 b = 6 Putting in the equation of ellipse conic section: x2/a2 + y2/b2 = 1 x2/64 + y2/36 = 1 |
Parabola
Parabola is a locus of a point that moves so that its distance from a fixed point is equal to the distance from the moving point to fixed straight lines. When the parabola has a focus at (a,0), with a > 0 and directrix x = -a, its equation can be written as:
y2 = 4ax.
Example of ParabolaQues. The equation of parabola is y2 = 40x. Find the length of the latus rectum, focus, and vertex. Ans. Length of latus rectum, focus and vertex of the parabola is as follows: Given: Equation of a parabola: y2 = 40x Therefore, 4a = 40 a = 40/4 = 10 Now, parabola formula for latus rectum is: Length of latus rectum = 4a = 4(10) = 40 Now, focus= (a,0) = (10,0) Now, Vertex = (0,0 |
Hyperbola
Hyperbola is the set of all points in a plane, the difference of whose distance from any two fixed points in the plane is constant. The equation of a hyperbola having its foci on the x-axis is:
(x−a)2/h2−(y−b)2/k2=1
Example of HyperbolaExample. What will be the equation for the hyperbola which has center at (1, 4), vertex at (2, 4), 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 = 4 Hence b2 = c2- a2 = 16 – 1 = 15. 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)2/15 = 1 |
Conic Sections Formulas
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The table below summarizes the conic section formulas used:
| Category | Formula | Description |
|---|---|---|
| Circle | (x−a)2 + (y−b)2 = r2 | where (a, b) is the center and r is the radius |
| Horizontal major axis ellipse | (x−a)2/h2 + (y−b)2/k2 = 1 | where (a,b) is the center, the length of the minor axis is 2k, the length of the major axis is 2h, distance between the centre and either of the focus is c with c2 = h2 − k2, h > k >0. |
| Vertical major axis ellipse | (x−a)2/k2+(y−b)2/h2=1 | where (a,b) is the center, length of the major axis is 2h, length of the minor axis is 2k, the distance between the centre and either of the focus is c with c2 = h2 − k2, h > k >0. |
| Horizontal transverse axis hyperbola | (x−a)2/h2−(y−b)2/k2=1 | where (a,b) is the center, distance between the vertices is 2h, distance between the foci is 2k, c2= h2 + k2. |
| Vertical transverse axis hyperbola | (x−a)2/k2−(y−b)2/h2=1 | where (a,b) is the center, distance between the vertices is 2h, distance between the foci is 2k, c2= h2 + k2 |
| Horizontal axis parabola | (y−b)2=4p(x−a), p≠0, | where (a,b) is the vertex, focus is (a+p,b), directrix is the line, x = a−p, axis is the line y=b |
| Vertical axis parabola | (x − a)2 = 4p(y−b), p≠0, | where (a,b) is the vertex, focus is (a+p, b), directrix is the line, x = b−p, axis is the line x=a. |
Read More: Equation for Hyperbola: Formula & Solved Examples
Identifying a Conic Section
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Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is the standard form of equation of a conic section, where the real numbers are: A, B, C, D, E, F with A ≠ 0, B ≠ 0, C ≠ 0.
- The conic section is a circle if A = C and B = 0 and eccentricity, i.e. e = 0
- It is an ellipse if B2– 4AC< 0.
- The section is a parabola if B2 – 4AC= 0.
- The section of a cone is a hyperbola if B2 – 4AC > 0, then.
- If 0<e<1, the conic is an ellipse
- If e = 1,it is a parabola
- In case of e>1, it is a hyperbola
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Things to Remember
- A conic section is defined as the locus of a point P moving in the plane of a fixed point F, known as focus.
- A fixed-line d known as directrix (with the focus not on d).
- The ratio of point P's distance from focus F to its distance from d equals a constant e known as eccentricity.
- The line touching the conic section externally at one point is called a tangent.
- A circle formed on the major axis of the ellipse diameter is known as an auxiliary circle.
- The point of intersections of the tangent drawn perpendicular to the ellipse is called the director circle.
Sample Questions
Ques. Find the equation of circle with center (-2, 3) and radius 4. (2 marks)
Ans. Here, h=-2, k=3, r=4
=> (x-h)2+ (y-k)2= r2
=> (x+2)2+ (y-3)2= 42
=> x2 + 4 + 4x+ y2+ 9- 6y= 16
=> x2 + y2+4x-6y-3=0
Ques If a circle has equation - (x+5)2 + (y-3)2 = 36, then find the center and radius of the circle. (2 marks)
Ans. The given equation is (x+5)2 + (y-3)2= 36
=> (x+5)2 + (y-3)2=62
=> (x-(-5))2 + (y-3)2=62
Therefore, h= -5, k=3 and r= 6
Hence the center lies at (-5,3) and the radius is 6.
Ques. Find the coordinates of the axis of the parabola y2 = 12x, its focus, the length of the latus rectum, and the equation of directrix. (3 marks)
Ans. The given equation is y2 = 12x. Here, the coefficient of x is positive. Hence, the parabola opens towards the right. When this equation is compared with y2 = 4ax, we get 4a = 12
⇒ a = 3
Hence, the Coordinates of the focus of the parabola(a, 0) = (3, 0)
As the given equation has y2, the x-axis is the axis of the parabola.
Equation of directrix, x = –a
i.e., x = – 3
i.e., x + 3 = 0
Length of latus rectum (4a) = 4 × 3
= 12
Ques. Find the coordinates of the foci, the vertice, the length of the minor and major axis, the length of the latus rectum, and the eccentricity, of the ellipse with the equation. (3 marks)
Ans. The solution is as follows:
Ques. What is the equation of a hyperbola that has its foci at (0,14) and passes through the point P(3,4). (4 marks)
Ans. The equation of the hyperbola
Let, the equation of the hyperbola be:
x2a2- y2b2=1
It passes through the point P(3,4)
Putting the values of (x,y)-
Ques. Find the equation of circle with center (-2, 4) and radius 6. (2 marks)
Ans. Here, h=-2, k=4, r=6
=> (x-h)2+ (y-k)2= r2
=> (x+2)2+ (y-4)2= 62
=> x2 + 4 + 4x+ y2+ 16- 8y= 36
=> x2 + y2+4x-8y-16=0
Ques If a circle has equation: (x+4)2 + (y-3)2 = 16, then find the center and radius of the circle. (2 marks)
Ans. The given equation is (x+4)2 + (y-3)2= 16
=> (x+4)2 + (y-3)2=42
=> (x-(-4))2 + (y-3)2=42
Therefore, h= -4, k=3 and r= 4
Hence the center lies at (-4,3) and the radius is 4.
Ques. What will be the equation for the hyperbola which has center at (1, 3), vertex at (2, 3), and the focus at (5, 3). (3 marks)
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 = 3
Hence
b2 = c2- a2 = 9 – 1 = 8.
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)2/8 = 1
Ques. If for an ellipse, the focus lies at (4, 0), a vertex lies at (5, 0), and its center lies at (0, 0). Find the equation of the ellipse. (2 marks)
Ans. From the given points, we can see that
c = 4 and a = 5.
Using b2 = a2 – c2
We get:
b2= 25 – 16 = 9
b = 3
Putting in the equation of ellipse conic section:
x2/a2 + y2/b2 = 1
x2/25 + y2/9 = 1
Ques. The equation of parabola is y2 = 36x. 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 = 36x
Therefore, 4a = 36
a = 36/4 = 9
Now, parabola formula for latus rectum is:
Length of latus rectum = 4a
= 4(9) = 36
Now, focus= (a,0) = (9,0)
Now, Vertex = (0,0)
Ques. Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = 6. (3 marks)
Ans. Given that, Focus = (0, 0) and directrix y = 6
Let us suppose that there is a point (x, y) on the parabola
The distance from the focus point (0, 0) is √((x − 0)2 + (y - 0)2 )
Its distance from directrix y = 6 is |y - 6|
By the definition of parabola, these two distances are the same.
√[(x − 0)2 + (y - 0)2] = |y - 6|
Squaring on both sides.
(x − 0)2 + (y - 0)2 = (y - 6)2
x2 + y2 = y2 - 12y + 36
x2 + 12y - 36 = 0
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