1509 circle- coordinate geometry
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The document covers coordinate geometry, focusing on the distance between points, the equations of lines and circles, and transformations of graphs. Key concepts include finding the gradient between points, determining perpendicular lines, and constructing equations for circles centered at various coordinates. It provides practical examples and formulas for calculating distances and shifts in graphs.
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f(x) + a is a shift in the x directionf(x) + a is a shift in the x direction
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1509 circle- coordinate geometry
- 1. C2 Coordinate geometryC2 Coordinate geometry From two pointsFrom two points Building the equation of a circleBuilding the equation of a circle Shifts of graphsShifts of graphs A circle centred at (a, b)A circle centred at (a, b)
- 2. Coord GeometryCoord Geometry From two pointsFrom two points
- 3. Plot two points A and B on an x-y plane What is the distance between them? What is the distance between them? A (4, -2) B (-5, 1) 9 3 AB2 = 32 + 92 AB = 3√10 AB = √ 90 Work in pencil
- 4. A (4, -2) B (-5, 1) Plot two points A and B on an x-y plane What is the equation of the line joining them? Gradient = dy /dx = -3 /9 or -1 /3 and we can use point A Y – Y1 = m (X – X1) ⇒ Y – (-2) = -1 /3 (X – 4) ⇒ 3Y + 6 = -X + 4 ⇒ X + 3Y + 2 = 0
- 5. A (4, -2) B (-5, 1) Now add in a third point! C (2, 2) What is the equation of the perpendicular line joining C to line AB? Gradient = 3 since m1m2 = -1 for perpendicular lines And we can use point C Y – Y1 = m (X – X1) ⇒ Y – 2 = 3 (X – 2) ⇒ 3X – Y - 4 = 0
- 6. A (4, -2) B (-5, 1) Where do the lines meet? C (2, 2) Expect horrible fractions! X + 3Y + 2 = 0 (1) 3X – Y - 4 = 0 (2) X + 3Y + 2 = 0 (1) Add 9X – 3Y - 12 = 0 (2)x3 10X - 10 = 0 X = 1 and Y = -1
- 7. Circle GeometryCircle Geometry Constructing the equation of a circleConstructing the equation of a circle
- 8. Start with a point. You can fix this to be an integer distance from (0,0) if you want an easier life! P1 (8, 6) What is the distance from the origin to your point? Find some other points that are the same distance from the origin. What maths are you using? What shape are you making? PYTHAGORAS to make a CIRCLE with radius 10! P2 (-8,6) P5 (0,-10) P3 (-10,0) P4 (√90, -√10)
- 9. P (x, y) But what about the equation? Start with a general point (x, y) We have x2 + y2 = radius2 Or radius x2 + y2 = r2 Have a play with autograph on the network to get some circles of different sizes.
- 10. Circle GeometryCircle Geometry Shifts of graphsShifts of graphs
- 11. Identify this transformation of f(x) = x2
- 12. Identify this transformation of f(x) = 1 /x
- 13. Identify this transformation of f(x) = cos x
- 14. Transformations of graphs - shiftsTransformations of graphs - shifts f(x + a) is a shift in the x directionf(x + a) is a shift in the x direction by vectorby vector -a 0[ ] 0 a[ ] f(x) + a is a shift in the x directionf(x) + a is a shift in the x direction by vectorby vector f(x) f(x)
- 15. Circle GeometryCircle Geometry The equation of a circle centre (a,b)The equation of a circle centre (a,b)
- 16. Draw a circle centred at (0, 0) with a whole number radius Work in pencil e.g. x2 + y2 = 102 10 10 Now shift the circle To the right e.g. by 3 What is the equation now? (x - 3)2 + y2 = 102 133
- 17. 10 Now shift the circle upwards e.g. by 4 What is the equation now? we have (x - 3)2 + y2 = 102 133 This is a bit trickier to understand. Ask your teacher for more explanation if you need it. (x - 3)2 + (y - 4)2 = 102 14 4 The centre of this circle is? (3,4) Now give a partner the point and radius you started with and check of they get the same answers.
- 18. Summary of resultsSummary of results You should know how to find the distanceYou should know how to find the distance between two points or the length of a given linebetween two points or the length of a given line You should know how to find the gradient andYou should know how to find the gradient and equation of a line given two pointsequation of a line given two points You should know about the gradient ofYou should know about the gradient of perpendicular lines and how to find the interceptperpendicular lines and how to find the intercept of two linesof two lines The equation of a circle centred at the origin isThe equation of a circle centred at the origin is xx22 + y+ y22 = r= r22 The equation of a circle centred at point (a, b) isThe equation of a circle centred at point (a, b) is (x – a)(x – a)22 + (y – b)+ (y – b)22 = r= r22
Editor's Notes
- #3: The animated gifs create immediate interest and can increase a pupil’s attention span
- #8: The animated gifs create immediate interest and can increase a pupil’s attention span
- #11: The animated gifs create immediate interest and can increase a pupil’s attention span
- #16: The animated gifs create immediate interest and can increase a pupil’s attention span