Chapter 10 Coordinate Treatment of Simple Locus Problems

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Path of a Moving Point (I) Given that P is a point on a wheel, what is the path of P when the wheel rolls on the ground?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Path of a Moving Point (II) Given that P is a vertex of a rectangle, What is the path of P if the rectangle rolls on the horizontal level?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Path of a Moving Point (III) Given that P is a vertex of a triangle, What is the path of P if the triangle rolls on the horizontal level?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Locus  A path traced by a moving point under given condition(s) is called a locus.  e.g. move point P such that  APB remains unchanged. What is the locus of P?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems What is the restriction of each drawing device shown on the left? Drawing Devices

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Collinear Points (I) Are points A, B and P collinear? If slope of AP  slope of PB, then A, P and B are collinear. ∵ Slope of AB , slope of BP  1 ∴ A, P and B are not collinear.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Collinear Points (II) Are points A, B and P collinear? If slope of AP  slope of PB, then A, P and B are collinear. ∵ Slope of AB  0.5, slope of BP  0.5 ∴ A, P and B are collinear.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems If P(x, y) is a moving point such that slope of BP equals slope of AB, what is the locus of P? How to find an equation connecting x and y? ∵ slope of AB  slope of BP ∴ P lies on the straight line which passes through A and B. Equations of Straight Lines ∵ slope of AB  0.5 and slope of BP  ∴ 0.5  i.e.x  2y  2  0 This kind of equation is called the equation of a straight line.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems  General form of a straight line: Ax  By  C  0 where A and B are not both zero.  Slope   x -intercept   y -intercept = General Form of Straight Lines

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems  If L 1 // L 2, then m 1  m 2.  If m 1  m 2, then L 1 // L 2.  If L 1  L 2, then m 1  m 2  1 。  If m 1  m 2  1, then L 1  L 2 。 General Form of Straight Lines

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Locus of the Tip What is the locus of the tip of the minute-hand? The locus is a circle!

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Equations of Circles OP  4 If the coordinates of the tip P of minute-hand are (x, y), what is the equation connecting x and y?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems Equations of Circles x 2  y 2  4 2 OP  4

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 10 Coordinate Treatment of Simple Locus Problems  The standard form of a circle with centre (h, k) and radius r: (x  h) 2  (y  k) 2  r 2  The general form of a circle x 2  y 2  Dx  Ey  F  0  By completing the square, Equations of Circles

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