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Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY : Analytical Geometry or Co-ordinate geometry is the branch of Mathematics
which deals with the study of geometry through algebra. It was Rene Descartes (1596-1650) a famous French Mathematician, who introduced algebraic methods to solve geometrical problems.

The entire subject is a progressive development of the basic idea of a “point”. Descartes established a relationship between the basic geometric concept of Point with basic algebraic entity Number. This relationship is called as system of co-ordinates. The plane in which the points are represented by an ordered pair (x,y) of real numbers uniquely and conversely is called the Cartesian Plane.

Axes of reference: Every ordered pair (x, y) of real numbers x and y can be represented by a point in a plane with reference to two fixed lines called
reference axes which may be mutually perpendicular or may be not perpendicular. If the two lines are perpendicular then the axes are called
Rectangular, otherwise they are called Oblique.

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

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:

I. Weightage to Objectives:

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

II. Weightage to level of difficulty:

Level Weightage Marks
Easy35%53/150
Average55%82/150
Difficult10%15/150

III. Weightage to content:

 Chapter No.   Chapter No. of teaching Hours  Marks
 1. SETS88
 2. RELATIONS AND FUNCTIONS1011
 3. TRIGONOMETRIC FUNCTIONS  18  19
 4. PRINCIPLE OF MATHEMATICAL INDUCTION 4  5
  5. COMPLEX NUMBERS AND QUADRATIC
EQUATIONS
 8 9
 6. LINEAR INEQUALITIES87
 7. PERMUTATION AND COMBINATION 99
 8. BINOMIAL THEOREM78
 9. SEQUENCE AND SERIES  9 11
10. STRAIGHT LINES 10  10
11. CONIC SECTIONS 99
12. INTRODUCTION TO 3D GEOMETRY 5 7
13. LIMITS AND DERIVATIVES 1415
14. MATHEMATICAL REASONING66
15. STATISTICS77
16. PROBABILITY89
  Total 150150

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY – Syllabus

1. Straight Lines

  • Brief recall of 2-D from earlier classes: mentioning formulae .
  • Slope of a line : Slope of line joining two points , problems
  • Angle between two lines: slopes of parallel and perpendicular lines, collinearity of three points and problems.
  • Various forms of equations of a line:
  • Derivation of equation of lines parallel to axes, point-slope form, slope-intercept form, two-point form, intercepts form and normal form and problems.
  • General equation of a line. Reducing ax+by+c=0 into other forms of equation of straight lines.
  • Equation of family of lines passing through the point of intersection of two lines and Problems.
  • Distance of a point from a line , distance between two parallel lines and problems.

2. 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.

3. Introduction to Three-dimensional Geometry

  • Coordinate axes and coordinate planes in three dimensions. Coordinates of a point.
  • Distance between two points and section formula and problems.

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

RECTANGULAR CARTESIAN COORDINATE SYSTEM

Draw two straight lines X’OB and Y’OY intersecting each other at right angles and let O be their point of intersection. Now X’OB is called the x-axis (axis
of x) and Y’OY is called the y-axis (axis of y). O is called the origin.

The two mutually perpendicular lines taken together are called as rectangular axes or coordinate axes or the axes of co-ordinates.

Let P be any point in the plane. Through P draw PM parallel to the y axis cutting the x axis at M and draw PN parallel to the x axis cutting the y axis at N as shown in the figure below :

Then OM is called the x co-ordinate or abscissa of the point P denoted by ‘x’ and MP is called the the y co-ordinate or ordinate of the point P denoted by ‘y’.

Then the point P is completely determined by the ordered pair (x,y) of real numbers.

This ordered pair (x,y) is the co-ordinates of P and this is written as P (x,y).

The above system of co-ordinating an ordered pair (x,y) with every point in a plane is called the Rectangular Cartesian Co ordinate system .

Note:

  • The co-ordinates of origin is taken as O(0,0)
  • Any point on x axis can be taken as (x,0) since for any point on the x axis, the y co-ordinate is zero.
  • Any point on y axis can be taken as (0,y) since for any point on the y axis, the x co-ordinate is zero
  • The abscissa of any point is numerically equal to the distance of the point from the y axis and the ordinate of any point is numerically equal
    to the distance of the point from the x axis.

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

Quadrants:

Two mutually perpendicular lines X’OX and Y’OY divide the plane into four parts called as the Quadrants. The region XOY is called the first Quadrant, the region YOX’ is called the second quadrant, the region X’OY’ is called the third quadrant and the region Y’OX is called the fourth quadrant.

According to our convention of signs of measurement of distances along the X and Y axes the following table gives the sign of x and y co-ordinates of
the points in different quadrants.

Quadrantx co-ordinatey co-ordinate
First ++
Second+
Third
Fourth+

Reflection / Image of a point with respect to the x axis and y axis

To find the image of a point A(1,2) in x axis , produce AM to A’ such that MA’ =2 units. We arrive at the point A'(1,-2) which is the image of A in the x axis.
Similarly to find the image of A(1,2) in y axis, produce AN to B’ such that NB’=1 unit. Then B'(-1,2) is the reflection or the image of the point A in the y axis.

Note: When a point is reflected in the x axis, the sign of its ordinate changes.

For eg, the image of the point A(1,2) in the x axis is A(1,-2).Thus the image of the point S(x,y) in x axis is S'(x,-y).When a point is reflected in the y axis the
sign of its abscissa changes. For example, the image of the point B(2,3) in the y axis is given by B'(-2,3)Thus the Image of the point A (x,y) in y axis is A'(-x,y) .

Reflection of a point in the origin

When a point P(x,y) is reflected in the origin the signs of its abscissa and ordinate both changes. Thus the reflection or the image of the point P(2,3) in the origin is given by P'(-2,-3).

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

DISTANCE BETWEEN TWO POINTS (DISTANCE FORMULA)

The distance between any two points in the plane is the length of the line segment joining them. The distance between two points P(x1,y1) and Q(x2,y2) is
given by 

|PQ| = √[(x2 – x1 )2 +(y2 – y1 )2] = √[(x1 – x2 )2 +(y1 – y2 )2]

Hence the distance PQ is given by

√[(difference of x coordinates)2 + (difference of y coordinates)2]

Note:

  • The distance of the point P(x,y) from the origin O(0,0) is given by  √[(x – 0 )2 +(y – 0 )2] = x2 + y2
  • When the line PQ is parallel to the y axis , the x co ordinates or the abscissa of the points P and Q will be equal and so the distance PQ is given by
    (y2 – y1) or (y1 – y2) and it is taken to be positive always.
  • When the line PQ is parallel to the x axis, the y co ordinates of the points P and Q will be equal and so the distance PQ is given by (x2 – x1) or (x1 – x2) and taken to be positive always.

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

Application of distance formula in geometrical problems

1. If A, B and C are any three given points in the plane, we have the following
results
a. If the sum of the distances between two line segment is equal to the length of the 3rd line segment, then the three points are said to be collinear

b. If any two sides of a triangle are equal then the three points form the vertices of an isosceles triangle

c. If all the three sides are equal then the three points form the vertices of an equilateral triangle

d. If the sum of the squares of lengths of any two sides of the triangle is equal to the square of the third side length (Pythagoras theorem) then the points form a right angled triangle.

2. If A,B,C and D are four points ,no three of which are collinear , the type of quadrilateral formed by these points is determined by using the distance
formula based on the properties:

i) Square →, prove that the four sides are equal and the diagonals are equal

ii) Rhombus → prove that the four sides are equal and the diagonals are not equal

iii) Rectangle → prove that opposite sides are equal and the diagonals are also equal

iv) Parallelogram → prove that opposite sides are equal and the diagonals are not equal

Karnataka Class 11 Commerce Maths UNIT III COORDINATE GEOMETRY

SECTION FORMULA

In this section we shall note a formula to find the co-ordinates of the point which divides the line joining the two points in the given ratio internally or
externally. Let AB be a straight line. If P is a point on the straight line AB, then AP/PB is called the position ratio of P on AB. If P lies between A and  B then P is said to divide AB internally in the given ratio. In this case as both AP and PB are measured in the same direction , they are of the same sign and hence the ratio AP:PB is positive.

On the other hand, if P lies on AB produced as shown in the figure, then the division is said to be external and here AP and PB are measured in the opposite directions and so they are of different signs and hence the ratio AP:PB is negative.

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