In simple terms, a conic section is a slice through a cone. It refers to the curves obtained when a plane surface intersects a double right circular cone. By altering the angle and the points of intersections, different types of curves can be produced.

The four basic types of curves are Circle, Ellipse, Hyperbola, and Parabola. Intersections do not pass through the vertices of the cone. A conic section can be easily graphed on a coordinate plane. All conic sections have some features, which include a focus and a directrix. Ellipses and hyperbolas have two foci and two directrix, while parabolas have one of each. In mathematical terms, a conic section is referred to as a set of points whose distance to the focus of the curve is a persistent multiple of their distance to the directrix.

Key Terms

Some key terms associated with conic sections are as follows:

• • Vertex: A point on the far-end of a conic section.

  • Locus: When a given equation or condition is satisfied by the coordinates of a set of points, this set of points is called locus.

  • Focus: The point at which the rays that are reflected from the curve converge. 

  • Nappe: Half part of a double cone; it resembles a party hat.

Applications of Conic Sections

Conic sections find their applications in many fields, mainly to define shapes. For instance, they have been used to explain the shapes of the objects’ orbits found in space in astronomy. As per Newton’s law of universal gravitation, two huge objects interacting with each other can move in conic section-shaped orbits in space. Depending on their characteristics, the orbits can be parabolas, hyperbolas, or ellipses.

Here in NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections, you will learn about the different types of conic sections.

When the plane that is intersecting the cone is parallel to its base, a circle is formed. A set of points, equidistant from the central axis of the cone is the place of intersection. Common features of all the circles are a center and a radius. Radius is basically the distance from any point on the circumference of the circle to the center.

The standard equation for a circle is: 

(x−h)² + (y−k)² = r²

Here, centre is (h, k) and radius is r. The radius has to be more than 0. If the radius is 0, it will be a degenerate case, where on the graph it is a point. When the perpendicular plane intersects at the tip of the cone, making a zero radius circle, it is referred to as the degenerate form of the circle.

A U-shaped curve formed by a parallel plane intersecting the cone is a parabola. Some features that are found in a parabola are:

• • Vertex: It is a point on the U-shaped curve from where it turns around.

  • Focus: This point lies in the interior of the curve, on the axis of symmetry.

  • Axis of Symmetry: The line dividing the parabola into equal halves and joining the focus and the vertex.

Standard equations of parabola

y2 = 4ax, is the standard equation of the parabola.

Latus rectum

It is a perpendicular line segment that passes through the focus of the parabola and runs parallel to the directrix. It touches the curve at both ends. In a parabola, it is always 4 times the length of its focal length. Therefore, the formula for the latus rectum is 4a, where ‘a’ is the focal length.

An ellipse is a set of points in a plane whose sum of distances from two points is a constant value. These two fixed points are referred to as the foci of the ellipse. In other terms, an ellipse is formed when the intersecting plane’s angle falls between the outside surface and the base of the cone. Some features common across all ellipses are as follows:

• • Major axis: The longest width across the ellipse.

  • Minor axis: The shortest width across the ellipse.

  • Centre: The intersection of the two axes

  • Foci or two focal points: The two points from which the sum of distances to any point on the ellipse is a constant value.

Relationship between semi-major axis, semi-minor axis and the distance of the focus from the center of the ellipse

a² = b² + c², where a is the semi-major axis, b is the semi-minor axis and c is the distance of the focus from the center of the ellipse.

Eccentricity

The eccentricity of the parabola is equal to 1. The formula is E= c/a, where E= Eccentricity, c= distance between the focal points and a= length of the major axis.

Standard Equation of Ellipse

The standard form of the equation of an ellipse with center (0,0) and major axis parallel to the x-axis is

X²/a² + y²/b² = 1

Latus rectum

It is the line segment that is perpendicular to the major axis. This major axis passes through the focus and its extreme point is on the ellipse. The equation for the latus rectum of the ellipse is 2b²/a.

The curve formed by the intersection of a plane that is perpendicular to the base of both cones, dividing them into two halves is called a hyperbola. A hyperbola has two branches along with other features like asymptote lines, a center, foci and vertices.

Eccentricity

The eccentricity of the hyperbola is always greater than 1. i.e., E > 1.

Standard equation of Hyperbola

(x − h) 2a²− (y − k) 2b² = 1 or (y − k) 2b²− (x − h) 2a² = 1

In this formula, a and b are the lengths of the semi-major and semi-minor axes, respectively.

Latus rectum

The line that is at a right angle to the transverse axis and passes through any of the two focal lengths with its end points lying on the hyperbola is the latus rectum. The formula is 2b²/4a, where a is the semi-major axis and b is the semi-minor axis.

Problem 1. Identify the conic section represented by the equation 2x²+2y²−4x−8y=40

Then graph the equation.

Answer: Circle.

Solution:

2x²+2y²−4x−8y=40⟹x²+y²−2x−4y=20

(x−1)²−1+(y−2)²−4=20

(x−1)²+(y−2)²=25

is a circle with radius of 5 and center (1,2)

Problem 2. Identify the conic section represented by the equation 4x²−4xy+y²−6=0

Answer: Parabola.

Solution:

4x²−4xy+y²−6=0

The general equation for any conic section is

Ax²+Bxy+Cy²+Dx+Ey+F=0

A is 4.

B is -4.

C is 1.

The discriminant is B² − 4AC= 16 – 4 (4)(1)=0 ⟹ is a parabola.

How to get full marks in Conic Sections?

To secure full marks in this chapter, you must first clear your basic concepts with the help of NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections. and then solve as many problems as possible. Go through the previous years’ question papers of CBSE exams and understand what kind of questions are asked. With these NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections, you can get to know the areas you are lagging behind in, and accordingly, you can work on them to ensure securing full marks in conic sections.

How many topics does Chapter 11 of NCERT Solutions for Class 11 Maths cover?

The NCERT Solutions for NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections include important topics like the circle, parabola, hyperbola, and ellipse. This chapter offers vital knowledge on different curves and their standard equations. It covers all the crucial topics that are often asked about in the Term-II exams. To be able to solve problems related to the conic section successfully, it is essential to practice them regularly.

Where can I get an accurate Conic Section Class 11 Solution?

MSVGo offers an accurate NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections. Since mathematics is all about accuracy, our team of experts consciously makes efforts to provide exact solutions for conic sections. These solutions have been formulated keeping in mind the type of questions usually asked in CBSE Term-II exams. Therefore, with regular practice and a thorough understanding of the concepts of conic sections, students can secure good marks.