1 LC.01.2 – The Concept of a Locus MCR3U - Santowski

2 (A) Skill Review Recall how to complete the square: Recall how to complete the square: 2x 2 – 12x + 1 2x 2 – 12x + 1 = 2(x 2 – 6x + 9 – 9) + 1 = 2(x 2 – 6x + 9 – 9) + 1 = 2(x – 3) 2 – 17 = 2(x – 3) 2 – 17 Recall simple binomial expansions  (a + b) 2 = a 2 + 2ab + b 2 which we now apply binomial expansion to equations with radicals Recall simple binomial expansions  (a + b) 2 = a 2 + 2ab + b 2 which we now apply binomial expansion to equations with radicals (6 -  x) 2 (6 -  x) 2 = (6) 2 + 2(6)(-  x) + (-  x) 2 = (6) 2 + 2(6)(-  x) + (-  x) 2 =  x + x =  x + x Recall how to find distances on a Cartesian plane  A(-5,3) and B(-1,-2) Recall how to find distances on a Cartesian plane  A(-5,3) and B(-1,-2)  ((3 - -1) 2 + ( ) 2 )  ((3 - -1) 2 + ( ) 2 )  (16 + 9) = 5  (16 + 9) = 5

3 (A) Skill Review Expand Expand

4 (B) Locus Definition A locus is a set of points that satisfy a given condition or conditions A locus is a set of points that satisfy a given condition or conditions For example, the line x = 2 could be defined as the set of all points that are 2 units to the right of the y-axis For example, the line x = 2 could be defined as the set of all points that are 2 units to the right of the y-axis For example, a circle could then be defined as a set of points that are equidistant from a fixed point (i.e. the center)  ex. The set of points that are 4 units from the origin would be the circle x 2 + y 2 = 4 2 For example, a circle could then be defined as a set of points that are equidistant from a fixed point (i.e. the center)  ex. The set of points that are 4 units from the origin would be the circle x 2 + y 2 = 4 2 For example, the set of points that are equidistant from two fixed points describes the line of the perpendicular bisector between the given two fixed points, say (-2,5) and (6,9) For example, the set of points that are equidistant from two fixed points describes the line of the perpendicular bisector between the given two fixed points, say (-2,5) and (6,9)

5 (C) Locus Definition - Diagrams

6

7 (D) In-Class Examples ex 1. Draw a diagram showing all points that are 2 units above the line y = 4 and determine the equation ex 1. Draw a diagram showing all points that are 2 units above the line y = 4 and determine the equation ex 2. Draw a diagram showing all points that are 3 units to the left of x = 1 and determine the equation ex 2. Draw a diagram showing all points that are 3 units to the left of x = 1 and determine the equation ex 3. Draw a diagram showing all points that are 3 units from the origin and determine the equation ex 3. Draw a diagram showing all points that are 3 units from the origin and determine the equation ex 4. Draw a diagram showing all points that are equidistant from (0,0) and (2,0) and determine the equation ex 4. Draw a diagram showing all points that are equidistant from (0,0) and (2,0) and determine the equation ex 5. Draw a diagram showing all points which start at (1,-2) and move up and right at angle of 45  and determine the equation ex 5. Draw a diagram showing all points which start at (1,-2) and move up and right at angle of 45  and determine the equation

8 (D) In-Class Examples ex 6. Draw a diagram showing all points that meet the following criteria: Point N has co-ordinates (1,-2) and Point P moves so that the slope of NP is always ¾ and determine the equation ex 6. Draw a diagram showing all points that meet the following criteria: Point N has co-ordinates (1,-2) and Point P moves so that the slope of NP is always ¾ and determine the equation ex 6. Draw a diagram showing all points which are equidistant from (2,0) and (0,3) and determine the equation ex 6. Draw a diagram showing all points which are equidistant from (2,0) and (0,3) and determine the equation ex 7. Draw all the points traced out by point P as it moves under the condition that the segment OP is perpendicular to PD if O(-4,0) and D(4,0) and determine the equation ex 7. Draw all the points traced out by point P as it moves under the condition that the segment OP is perpendicular to PD if O(-4,0) and D(4,0) and determine the equation

9 (E) Homework AW text, page459, Q1,7,8,9,14,15 AW text, page459, Q1,7,8,9,14,15 Nelson text, p574, Q1,2,4,5,11,13,14 Nelson text, p574, Q1,2,4,5,11,13,14