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Math1.1

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  • 1. Topics : Chapter 1 : Conic section Chapter 2 : Differential Equation Chapter 3 : Numerical Method Chapter 4 : Data descriptions Chapter 5 : Probability and Random variables Chapter 6 : Special Distribution Function Mathematics 102/3
  • 2. ASSESSMENT 100% Total 100 2 hours Subjective Question All topic Paper 2 60% 100 2 hours Subjective Question All topic Paper 1 Final examination 20% - Throughout The semester Assessment/Quiz/ Tutorial - - Continuous Assessment 20% 100 2 Hour Subjective Question - 1 Test Percentage Marks Time Format Topic Paper Components
  • 3. MAT 102/3 CHAPTER 1: CONIC SECTIONS
  • 4. 1.1 : Intoduction to conic sections
    • Circles, parabola, ellipses and
    • hyperbolas are called conic
    • sections because they are curves
    • obtained by the intersection of a
    • right circular cone and a plane.
    • The curves formed depends on
    • the angel at which the plane
    • intersects the cone.
  • 5.  
  • 6.  
  • 7. 1.2 : Circles Definition: A circle is defined as the set of all points P in a plane that are at a constant distance from a fixed point. This fixed point is called the centre and the fixed distance is called the radius
  • 8. y x P( x, y ) C ( h, k ) r Figure shows a circle with center (h,k) and radius r r X - h Y - k
  • 9. Equations of a cirles
    • From the definition of the circle, a point P(x,y) lies on the circle if and only if PC = r, that is
    Squaring both sides, we have This is the equation of the circle with center (a,b) and radius r units If the origin is the center of the circle, the equation becomes
  • 10.
    • Example 1
    • Find the equation of the circle with :-
    • Center at origin and radius 3 units
    • (ii) Center (2,-3) and radius 5 units
    • Solutions :
    (i) (ii)
  • 11. General Equation of a Circle The equation of a circle with center (a,b) and radius r units is Now substituting g = a, f = k and c=a 2 +b 2 -r 2 Conversely, the equation Where g,f and c are constant, represent a circle This equation is called the general equation of a circle
  • 12. Center and radius of a circle x 2 +y 2 +2gx+2fy+c=0 Completing the squares for x 2 +2gx and y 2 +2fy, x 2 +y 2 +2gx+2fy+c=0 X 2 +2gx+g 2 +y 2 +2fy+f 2 =g 2 +f 2 -c (x+g) 2 +(y+f) 2 =g 2 +f 2 -c Hence, the center of a circle is (-g,-f) and the radius is
  • 13. Dertermine the equation with center (h,k) Example 2 Find the center and the radius of the circle x 2 +y 2 +5x-6y-5=0 Comparing with the general equation, x 2 +y 2 +2gx+2fy+c=0 g=5/2 f=-3 c=-5 Hence,the center is (-5/2,3) and the radius is
  • 14. Determine the centre and radius of a circle.
    • Example 3
    • Graph
    • Solution
    • We can change the given equation into the standard form of the circle by completing the square on x and y as follow
    4
  • 15. The center is at ( 3 , -2), and the length of a radius is 2 units y x (3,-2) r =2
  • 16. Point of Intersection
    • Example 4
    • Find the coordinates of the point of intersection of the circles x 2 +y 2 -4=0 and x 2 +y 2 -2x+4y+4=0
    • x 2 +y 2 -4=0 ……….(1)
    • x 2 +y 2 -2x+4y+4=0…..(2)
    • Solving the equation simultaneously for the point of intersection,
    • – (2) 2x-4y-8=0
          • x=2y+4
  • 17. Substituting x=2y+4 in (1) gives (2y+4) 2 +y2-4=0 4y 2 +16y+16+y2-4=0 5y 2 +16y+12=0 (5y+6)(y+2)=0 Y=-6/5 or 2 When y=-6/5, x=12/5+4=8/5 When y=-2 x=-4+4=0 Therefore, the points of intersection are (8/5,-6/5) and (0,-2)
  • 18. Point of a circle and a straight line Example 5 Find the coordinates of the points of intersection between the circle X 2 +y 2 -6x+9=0 and the line y=7-x Solution:- Given X 2 +y 2 -6x+9=0….(1) y=7-x….(2) By substituting (2) into (1) gives, on simplication X 2 -8x+15=0 (x-5)(x-3)=0 x=5, y=2 x=3, y=4 So, intersection point are (5,2) and (3,4)
  • 19. Circle passing through three given points
    • If we are given the coordinates of three points on the circumference of a circle, we can substitute these values of x and y into the equation of the circle and obtain three equations which can be solved simultaneously to find the constants g, f and c .
  • 20.
    • Find the equation of the circle passing through the points (0,1). (4,3), and (1,-1).
    • Solution:
    Suppose the equation of the circle is points into this equation Substituting the coordinates of each of the three equation gives : ---------(1) --------(2) Circle passing through three given points 2+2g-2f+c+0 --------(3)
  • 21.
    • Multiplying equation [3] by 4 and then subtracting from equation [2], gives
    • Multiplying equation [1] by 3 and adding to equation [4] gives
    or Then from equation [1] And from equation [3] The equation of the circle which passes through (0,1), (4,3) and is
  • 22. Find the equation of a circle passing through two points with the equation of the diameter given
    • Example 6
    • Find the equation of the circle passing through the points (1,1) and (3,2) and with diameter
    Solution The standard form of the circle is Since the circle passes through -------[1] -------[ 2] Since the circle passes through
  • 23.
    • The center of the circle must passes through the diameter
    Therefore, -----[3] Solving equations [1], [2] and [3], given , and The equation of the circle is
  • 24. Tangent To A Circle
    • Teorem 1
    • Suppose we have a standard equation,
    • so the equation of a tangent for the circle at the point of is given by
    • see figure 1.2
    y x Figure 1.2
  • 25.
    • Find the equation of the tangent to a circle at the point
    Solution Method 1 By using the common tangent equation In this case and y = 3 . So the tangent is . Example 7
  • 26. Method 2 differentiating with respect to At the point gradient of tangent is .
  • 27. The length of the tangent to a circle
    • Teorem
    • The length of the tangent from a fixed point to a circle with equation
    • (denote by d ), is given by
  • 28. see figure 1.3 Figure 1.3
  • 29.
    • Find the length of the tangent from the point to the circle
    Solution We see that and By substituting this value in the equation d = ,we find = Example 8