Co ordinate Geometry ix

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Smt. T.S.Popat Head Mistress Sanghavi K.M.High School Gujarati Medium Co-ordinator B.O.S. MATHS

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Co-ordinate Geometry Chapter No. (G, 6) 1 4 5 3 2 7 9 8 6 1 5 4 3 2 6 7

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René Descartes "I think, therefore I am." Founder of Analytic Geometry Cartesian philosophy, was based on skepticism use of reason through logical analysis

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Geometry& Algebra United for the First time This is the Real beauty of the Topic

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Yaam Bhumiti Nirdeshak Bhumiti

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René Des ‘cartes’ “Cartes”ian Co Ordinate System Geometry and the Fly One morning Descartes noticed a fly walking across the ceiling of his bedroom. As he watched the fly, Descartes began to think of how the fly's path could be described without actually tracing its path. His further reflections about describing a path by means of mathematics led to La Géometrie  and Descartes's invention of coordinate geometry.

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cogito ergo sum I think therefore ,I am., Rene Descartes – without whom – we may not bedoing Coordinate Geometry!

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Algebraic Equation in Geometry x – 2y = 1 Line X—2y = 1 is a line in Geometry Geogebra

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After 2000 years of Euclidean Geometry This was the FIRST significant development by RENE DESCARTES ( French) in 17th Century, Part of the credit goes to Pierre Farmat’s (French) pioneering work in analytic geometry In his manuscript "Varia opera mathematica", ("Introduction to Plane and Solid Loci"). Several decades after Descartes Sir Isaac Newton (1640–1727) developed ten different coordinate systems. Newton and Leibnitz used the polar coordinate system  It was Swiss mathematician Jakob Bernoulli (1654–1705) who first used a polar co-ordinate system for  calculus problems and coined the terms"pole" and "polar axis” Turning point in the History of Mathematics

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Polar coordinate system Parabolic coordinate system Bipolar coordinates Hyperbolic coordinates Elliptic coordinates Cylindrical coordinate system Spherical coordinate system Parabolic coordinate system Parabolic cylindrical coordinates Paraboloidal coordinates Oblate spheroidal coordinates Prolate spheroidal coordinates Ellipsoidal coordinates Elliptic cylindrical coordinates Toroidal coordinates Bispherical coordinates Bipolar cylindrical coordinates Conical coordinates Flat-ring cyclide coordinates Flat-disk cyclide coordinates Bi-cyclide coordinates Cap-cyclide coordinates Variety!!! Development of Maths

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E x t r a D o s e

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Two intersecting line determine a plane. Two intersecting Number lines determine a Co-ordinate Plane/system. or Cartesian Plane. or Rectangular Co-ordinate system. or Two Dimentional orthogonal Co-ordinate System or XY-Plane ┴ GRID

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Use of Co-ordinate Geometry Cell Address is (D,3) or D3

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Use of Co-ordinate Geometry

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Use of Co-ordinate Geometry

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Use of Co-ordinate Geometry

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Use of Co-ordinate Geometry

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Use of Co-ordinate Geometry R A D A R MAP R A D A R

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Use of Co-ordinate Geometry Pixels in Digital Photos Each Pixel uses x-y co-ordinates

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The screen you are looking at is a grid of thousands of tiny dots called pixels that together make up the image

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Practical Application: public double distance(Point p, Point q) { double dx = p.x - q.x; //horizontal difference double dy = p.y - q.y; //vertical difference double dist = Math.sqrt( dx*dx + dy*dy ); //distance using Pythagoras theorem return dist;} All computer programs written in Java language, uses distance between two points.

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Lettering with Grid

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Terms Horizontal Vertical Above X-Axis Below X-Axis Right of Y-axis Left of Y-axis Half Plane origin Abscissa Ordinate Ordered Pair Quadrants Sign –Convention Frame of reference I IV II III

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A B C

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New topic Upgradation To Unit Shifted From New topic Area Of Triangle

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Introduction to Analytic Geometry

Dimensions : 

Dimensions 1-D 2-D 3-D

1-D : 

1-D | b-a | or | a-b | Distance Formula

2-D: “THE” Distance formula : 

2-D: “THE” Distance formula A B

2-D: “THE” Distance formula : 

2-D: “THE” Distance formula A B

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From 3D to 2D

Distance between two points.In general, : 

Distance between two points.In general, x1 x2 y1 y2 A(x1,y1) B(x2,y2) Length = x2 – x1 Length = y2 – y1 AB2 = (y2-y1)2 + (x2-x1)2 Hence, the formula for Length of AB or Distance between A and B is y x

Distance between two points. : 

Distance between two points. 5 18 3 17 A(5,3) B(18,17) 18 – 5 = 13 units 17 – 3 = 14 units AB2 = 132 + 142 Using Pythagoras’ Theorem, AB2 = (18 - 5)2 + (17 - 3)2 y x A ( 5 , 3 ) , B ( 18, 17 ) A ( x1 , y1 ) B ( x2 , y2 ) y2 - y1 = 17-3 X2 - x1 = 18-5

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Distance formula is nothing but Pythagoras Theorem A B

The mid-point of two points. : 

The mid-point of two points. x1 x2 y1 A(5,3) B(18,17) Look at it’s horizontal length Look at it’s vertical length Mid-point of AB y x y2 Formula for mid-point is

The mid-point of two points. : 

The mid-point of two points. 5 18 3 17 A(5,3) B(18,17) Look at it’s horizontal length = 11.5 Look at it’s vertical length = 10 (11.5, 10) Mid-point of AB y x (18,3)

Find the distance between the points (-1,3) and (2,-6) : 

Find the distance between the points (-1,3) and (2,-6) (-1, 3) (2, -6) (x1 , y1 ) (x2 ,y2 ) AB= 9.49 units (3 sig. fig) y2—y1= -6-3= -9 x2—x1=2--(--1)= 3

Topic Newly Introduced : 

Topic Newly Introduced Area of triangle when three vertices are given.

Area : 

Area Area of a Polygon. Three points A(Δ) = ½ [x1(y2-y3)+ x2 (y3-y1)+ x3 (y1-y2)] The area of triangle ABC is given by   This formula may be extended to a n sided polygon with n vertices. The area is then given by

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3 points are collinear if AB+BC=AC then A-B-C. Collinear Points Show that the points A(-5,4) , B(-2,-2) , C(3 ,-12) are Collinear Points. If (3,7),(2,5),(-2,-3) are the Vertices of a triangle.Show that these points are Collinear Points.

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Revision of all topics through Co-ordinate Geometry All types of Triangles and Quadrilaterals. Collinearity Area of Plane figures. and C,G,I,O etc…

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Lesson Plan periods

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Long Answer Question:- Sample Question 1 : Three vertices of a rectangle are (3, 2), (– 4, 2) and (– 4, 5). Plot these points and find the coordinates of the fourth vertex.

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Time Management Matrix

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Human Function Curve

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Stress arises when: Demands > Resources ...Think ,Plan ,Execute and BE HAPPY