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# Karnataka Class 11 Commerce Maths Conic Section

## Karnataka Class 11 Commerce Maths Conic Section

### Karnataka Class 11 Commerce Maths Conic Section

Karnataka Class 11 Commerce Maths Conic Section : Geometry is one of the most ancient branches of mathematics. The Greek geometers investigated the properties of many curves that have theoretical and practical importance. Euclid wrote his treatise on geometry around 300 B.C. He was the first who organised the geometric figures based on certain axioms suggested by physical considerations.

Geometry as initially studied by the ancient Indians and Greeks, who made essentially no use of the process of algebra. The synthetic approach to the subject of geometry as given by Euclid and in Sulbasutras, etc., was continued for some 1300 years. In the 200 B.C., Apollonius wrote a book called ‘The Conic’ which was all about conic sections with many important discoveries that have remained unsurpassed for eighteen centuries.

### Karnataka Class 11 Commerce Maths Conic Section

Karnataka Class 11 Commerce Maths Conic Section – SYLLABUS – Conic Section

Sections of a cone: Definition of a conic and definitions of Circle, parabola, ellipse, hyperbola as a conic .

Derivation of Standard equations of circle , parabola, ellipse and hyperbola and problems based on standard forms only.

### Karnataka Class 11 Commerce Maths Conic Section

Design of the Question Paper

MATHEMATICS CLASS : I PUC

Time: 3 hours 15 minute; Max. Mark:100

The weightage of the distribution of marks over different dimensions of the question paper shall be as follows:

### Karnataka Class 11 Commerce Maths Conic Section

I. Weightage to Objectives:

 Objective Weightage Marks Knowledge 40% 60/150 Understanding 30% 45/150 Application 20% 30/150 Skill 10% 15/150

### Karnataka Class 11 Commerce Maths Conic Section

II. Weightage to level of difficulty:

 Level Weightage Marks Easy 35% 53/150 Average 55% 82/150 Difficult 10% 15/150

### Karnataka Class 11 Commerce Maths Conic Section

III. Weightage to content:

 Chapter No. Chapter No. of teaching Hours Marks 1. SETS 8 8 2. RELATIONS AND FUNCTIONS 10 11 3. TRIGONOMETRIC FUNCTIONS 18 19 4. PRINCIPLE OF MATHEMATICAL INDUCTION 4 5 5. COMPLEX NUMBERS AND QUADRATIC EQUATIONS 8 9 6. LINEAR INEQUALITIES 8 7 7. PERMUTATION AND COMBINATION 9 9 8. BINOMIAL THEOREM 7 8 9. SEQUENCE AND SERIES 9 11 10. STRAIGHT LINES 10 10 11. CONIC SECTIONS 9 9 12. INTRODUCTION TO 3D GEOMETRY 5 7 13. LIMITS AND DERIVATIVES 14 15 14. MATHEMATICAL REASONING 6 6 15. STATISTICS 7 7 16. PROBABILITY 8 9 Total 150 150

### Karnataka Class 11 Commerce Maths Conic Section

IV. Pattern of the question paper:

 PART Type of questions Number of questions to be set Number of questions to be answered Remarks A 1  mark questions 10 10 Compulsory part B 2  mark questions 14 10 —————— C 3  mark questions 14 10 —————— D 5  mark questions 10 6 Questions must be asked from the specific set of topics as mentioned below, under section V E 10  mark questions(Each question with two subdivisions namely)a) 6 mark andb) 4 mark. 2 1

### Karnataka Class 11 Commerce Maths Conic Section

GUIDELINES TO THE QUESTION PAPER SETTER

1. The question paper must be prepared based on the individual blue print without changing the weightage of marks fixed for each chapter.

2. The question paper pattern provided should be adhered to.

• Part A : 10 compulsory questions each carrying 1 mark;
• Part B : 10 questions to be answered out of 14 questions each carrying 2 mark ;
• Part C : 10 questions to be answered out of 14 questions each carrying 3 mark;
• Part D : 6 questions to be answered out of 10 questions each carrying 5 mark
• Part E : 1 question to be answered out of 2 questions each carrying 10 mark with subdivisions (a) and (b) of 6 mark and 4 mark respectively. (The questions for PART D and PART E should be taken from the content areas as explained under section V in the design of the question paper)

3. There is nothing like a single blue print for all the question papers to be set. The paper setter should prepare a blue print of his own and set the paper accordingly without changing the weightage of marks given for each chapter.

4. Position of the questions from a particular topic is immaterial.

5. In case of the problems, only the problems based on the concepts and exercises discussed in the text book (prescribed by the Department of Pre-university education) can be asked. Concepts and exercises different from text book given in Exemplar text book should not be taken. Question paper must be within the frame work of prescribed text book and should be adhered to weightage to different topics and guidelines.

6. No question should be asked from the historical notes and appendices given in the text book.

7. Supplementary material given in the text book is also a part of the syllabus.

8. Questions should not be split into subdivisions. No provision for internal choice question in any part of the question paper.

9. Questions should be clear, unambiguous and free from grammatical errors. All unwanted data in the questions should be avoided.

10. Instruction to use the graph sheet for the question on LINEAR PROGRAMMING in PART E should be given in the question paper.

11. Repetition of the same concept, law, fact etc., which generate the same answer in different parts of the question paper should be avoided.

### Karnataka Class 11 Commerce Maths Conic Section

In this Chapter the following concepts and generalisations are studied.

• A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.
• The equation of a circle with centre (h, k) and the radius r is (x – h)2 + (y – k)2 = r2 .
• A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.
• The equation of the parabola with focus at (a, 0) a > 0 and directrix  x = – a is y2 = 4ax.
• Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.
• Length of the latus rectum of the parabola y2 = 4ax is 4a.
• An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
• The equation of an ellipse with foci on the x-axis is  x2 /a2 + y2 / b= 1.
• Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse.
• Length of the latus rectum of the ellipse  x2 /a2 + y2 / b= 1 is 2b2 / a.
• The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.
• A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.
• The equation of a hyperbola with foci on the x-axis is : x2 /a2 – y2 / b= 1.
• Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.
• Length of the latus rectum of the hyperbola : x2 /a2 – y2 / b= 1 is 2b2 / a.
• The eccentricity of a hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola.

### Karnataka Class 11 Commerce Maths Conic Section

Sections of a Cone

Karnataka Class 11 Commerce Maths Conic Section

### Karnataka Class 11 Commerce Maths Conic Section

Let l be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle α. Suppose we rotate the line m around the line l in such a way that the angle α remains constant. Then the surface generated is a double-napped right circular hollow cone herein after referred as cone and extending indefinitely far in both directions. The point V is called the vertex; the line l is the axis of the cone.

The rotating line m is called a generator of the cone. The vertex separates the cone into two parts called nappes. If we take the intersection of a plane with a cone, the section so obtained is called a conic section. Thus, conic sections are the curves obtained by intersecting a right circular cone by a plane.

We obtain different kinds of conic sections depending on the position of the intersecting plane with respect to the cone and by the angle made by it with the vertical axis of the cone. Let β be the angle made by the intersecting plane with the vertical axis of the cone. The intersection of the plane with the cone can take place either at the vertex of the cone or at any other part of the nappe either below or above the vertex.

Circle, ellipse, parabola and hyperbola

When the plane cuts the nappe (other than the vertex) of the cone, we have the following situations:

(a) When β = 90o , the section is a circle.

(b) When α < β < 90o , the section is an ellipse.

(c) When β = α; the section is a parabola. (In each of the above three situations, the plane cuts entirely across one nappe of the cone).

(d) When 0 ≤ β < α; the plane cuts through both the nappes and the curves of intersection is a hyperbola.

Degenerated conic sections

When the plane cuts at the vertex of the cone, we have the following different cases:

(a) When α < β ≤ 90o , then the section is a point.

(b) When β = α, the plane contains a generator of the cone and the section is a straight line. It is the degenerated case of a parabola.

(c) When 0 ≤ β < α, the section is a pair of intersecting straight lines. It is the degenerated case of a hyperbola.

### Karnataka Class 11 Commerce Maths Conic Section

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