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Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced PDF Download

Introduction

  • A curve, generated by intersecting a right circular cone with a plane is termed as ‘conic’. 
  • It has distinguished properties in Euclidean geometry. 
  • The vertex of the cone divides it into two nappes referred to as the upper nappe and the lower nappe.
    Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • The initials as mentioned in the above figure A carry the following meaning,
    (i) V is the vertex of the cone.
    (ii) l is the axis of the cone.
    (iii) m, the rotating line is a generator of the cone.
  • In figure B, the cone is intersected by a plane and the section so obtained is known as a conic section.
    Depending upon the position of the plane which intersects the cone and the angle of intersection β, different types of conic sections are obtained. i.e.
    (i) Circle
    (ii) Ellipse
    (iii) Parabola
    (iv) Hyperbola

Terminology

A conic section or conic is the locus of a point that moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.

  • The fixed point is called the Focus.
  • The fixed straight line is called the Directrix.
  • The constant ratio is called the Eccentricity denoted by e.
  • The line passing through the focus & perpendicular to the directrix is called the Axis.
  • A point of intersection of a conic with its axis is called a Vertex.

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

  • Eccentricity is the factor related to conic sections which show how circular the conic sections.
    More the eccentricity less circular the shape is. The eccentricity of the line is ∞.
    The two conic sections will be of the same shape if they have the same eccentricity.
    Suppose, the angle formed between the surface of the cone and its axis is β and the angle formed between the cutting plane and the axis is α, then
    eccentricity, e = cos α/cos β

Note:

(i) If eccentricity, e = 0, the conic is a circle

(ii) If 0<e<1, the conic is an ellipse

(iii) If e=1, the conic is a parabola

(iv) If e>1, it is a hyperbolaOverview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

  • Principal Axis: Line joining the two focal points or foci of ellipse or hyperbola. Its midpoint is the center of the curve.
  • Linear Eccentricity: Distance between the focus and center of a section.
  • Latus Rectum: A chord of section parallel to directrix, which passes through a focus.
  • Focal Parameter: Distance from focus to the corresponding directrix.
  • Major axis: Chord joining the two vertices. It is the longest chord of an ellipse.
  • Minor axis: Shortest chord of an ellipse.

Question for Overview: Conic Sections
Try yourself:
Which conic section has an eccentricity of 0?
View Solution

Circle

If β=90o, the conic section formed is a circle as shown below.
Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

The standard form of a circle: (x - h)2 + (y - k)2 = r2

where (h, k) is the center of the circle and r is the radius of the circle.Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

Ellipse

If α<β<90o, the conic section so formed is an ellipse as shown in the figure below.

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

The standard form of an ellipse is:

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

where (h, k) is the center of the ellipse, a is the horizontal stretch factor and b is the vertical stretch factor.

Overview: Conic Sections | Mathematics (Maths) for JEE Main & AdvancedEllipse with horizontal major axis: 

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

  • Focus: There are 2 focii; (ae, 0) and (-ae, 0)
  • Directrix: These foci have corresponding directrices asOverview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Length of major axis(xx’): 2a
    Length of major axis(yy’): 2b
  • Vertex: (a, 0) & (-a, 0)
  • Centre: (0, 0)
  • Latus rectum:Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Length of Latus rectumOverview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

Ellipse with vertical major axis:

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

  • Focii: (0, be) & (0, -be)
  • Directrices: x = b/e & x = -b/e
  • Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Length of major axis (yy'): 2b
    Length of minor axis (xx'): 2a
  • Vertex: (0, b) & (0, -b)
  • Centre: (0, 0)
  • Latus rectum:Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Length of Latus rectumOverview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

Note:

(i) Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
(ii) Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
(iii) Distance between 2 directices=Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
(iv) Distance between 2 focii = (major axis) × eccentricity
(v) Distance between focus and directrix =Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

Question for Overview: Conic Sections
Try yourself:
If ? = 60o and ? = 75o, what type of conic section is formed?
View Solution

Parabola

If α = β, the conic section formed is a parabola (represented by the orange curve) as shown below.
Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

The standard form of an ellipse is:
x = a(y - k)2 + h (east to west), where a is the horizontal stretch factor and (h, k) is the vertex.

y = a(x - h)2 + k (north to south), where a is the vertical stretch factor and (h, k) is the vertex.Overview: Conic Sections | Mathematics (Maths) for JEE Main & AdvancedStandard forms of Parabola

  • Latus rectum: 4 x (distance between vertex and focus) = 4a
  • Latus rectum: 2 x Distance between directrix and focus = 2(2a).
  • Point of intersection of Axis and directrix and the focus is bisected by the vertex.
  • Focus is the midpoint of the Latus rectum.
  • (Distance of any point on parabola from the axis)2 = (LR) (Distance of the same point from tangent at the vertex)

Illustration 1: If extreme points of LR are (11/2, 6) and (13/2, 4). Find the equation of the parabola.
Solution: Mid point of LR = focus = (6, 5)

⇒ 4a = 2 or a = ½

The equation of parabolas are: (y – 5)2 = 2(x – 5.5) and (y – 5)2 = – 2 (x – 6.5)

Question for Overview: Conic Sections
Try yourself:
If the extreme points of the Latus rectum are (7/2, 4) and (9/2, 6), what is the equation of the parabola?
View Solution

Hyperbola

If 0 ≤ β < α, then the plane intersects both nappes and the conic section so formed is known as a hyperbola (represented by the orange curves).
Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

The standard form of an ellipse is:
Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

where a is the horizontal stretch factor and b is the vertical stretch factor.

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

Standard Hyperbola

  • Equation: Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Focus: There are 2 focii (ae, 0) and (-ae, 0)
  • Directrix: The foci has corresponding directrices as x = +a/e and x = −a/e respectively.
  • Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Length of Transverse axis (xx') = 2a
    Length of Conjugate axis (yy') = 2b (Hypothetical)
  • Vertex: (0, 0) and (-a, 0)
  • Centre: (0, 0)
  • Latus rectum: x = +ae
  • Length of latus rectumOverview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
  • Position of point at hyperbola Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced.
    Let Overview: Conic Sections | Mathematics (Maths) for JEE Main & AdvancedandOverview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
    (i) If S1 > 0, point (x1, y1) lies inside the hyperbola
    (ii) If S1 = 0, point (x1, y1) lies on the hyperbola
    (iii) If S1 < 0, point (x1, y1) lies outside the hyperbola

Illustration 2: Classify the following equations according to their type of conics.

(a) 4x2 + 4y2 - 16x + 4y - 60 = 0

(b) x2 - 4x + 16y + 17 = 0

(c) x2 + 2y2 + 4x + 2y – 27 = 0

(d) x2 – y2 + 3x – 2y – 43 = 0

Solution:

(a) Here A = 4, B = 0, C = 4

Determinant will be B2 - 4AC = 02- 4(4) (4) = -64

This shows that B2 - 4AC < 0, B = 0 and A = C, so this is a circle.

In another way,
Since Both variables are squared, and both squared terms are multiplied by the same number. Therefore, this is a circle

(b) Here A =1,B = 0, C = 0

Determinant will be B2- 4AC = 0- 4(1) (0) = 0

This shows that B2 - 4AC = 0, so this is a parabola.
In another way,
Since only one of the variables is squared. Therefore, this is called a Parabola.

(c) Here A =1, B = 0, C = 2

Determinant will be B2 - 4AC = 02 - 4(1) (2) = -8

This shows that B2 - 4AC < 0, and A ≠ C so this is an ellipse.
In other way,
Since both variables are squared with the same sign, but they aren't multiplied by the same number. Therefore, so this is an ellipse.
(d) Here A =1,B = 0, C = -1

Determinant will be B2- 4AC = 02 - 4(1) (-1) = 8

This shows that B2 - 4AC > 0 so this is a hyperbola.
In another way,
Since both variables are squared, and the squared terms have opposite signs. Therefore, this is a hyperbola.

Question for Overview: Conic Sections
Try yourself:
Which conic section is represented by the equation x^2 - 4x + 16y + 17 = 0?
View Solution

Degenerated Conics

A degenerate conic is generated when a plane intersects the vertex of the cone.
There are three types of degenerate conics:

  1. If α< β≤90°, then the plane intersects the vertex exactly at a point. This condition is known as the degenerated form of circle or ellipse.

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

  1. If α=β, the plane upon an intersection with a cone forms a straight line containing a generator of the cone. This condition is a degenerated form of a parabola.

Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced

  1. If 0≤β<α, the section formed is a pair of intersecting straight lines. This condition is a degenerated form of a hyperbola.Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced
The document Overview: Conic Sections | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on Overview: Conic Sections - Mathematics (Maths) for JEE Main & Advanced

1. What are the key conic sections covered in the JEE syllabus?
Ans. The key conic sections covered in the JEE syllabus are circle, ellipse, parabola, and hyperbola.
2. How are circle and ellipse different from each other in terms of their equations and properties?
Ans. A circle has a constant distance from its center to any point on the circumference, while an ellipse has two different distances from its center to any point on the ellipse. The equations for a circle and ellipse also differ, with the general equation for a circle being (x-h)^2 + (y-k)^2 = r^2 and for an ellipse being (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
3. What is a degenerated conic in conic sections?
Ans. A degenerated conic is a special case where a conic section loses its defining characteristics and transforms into a simpler shape. For example, a degenerated conic can occur when the eccentricity of an ellipse becomes 0, resulting in a circle.
4. How are parabolas and hyperbolas distinguished from each other in conic sections?
Ans. Parabolas have a single focus point and a directrix, while hyperbolas have two focus points and two branches. The general equation for a parabola is y^2 = 4ax or x^2 = 4ay, while for a hyperbola, it is (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/a^2 - (x-h)^2/b^2 = 1.
5. How are conic sections used in real-life applications?
Ans. Conic sections have various real-life applications, such as in astronomy for describing the orbits of planets, in optics for designing lenses and mirrors, in architecture for creating domes and arches, and in engineering for designing antennas and satellite dishes.
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