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ANALYTIC GEOMETRY - Coggle Diagram

- - - - If P is a point which is an element of A = {(x,y) I x.y > 0 and x,y ∈ R}, P lies in first or third quadrant.
      - If P is a point which is an element of B = {(x,y) I x.y < 0 and x,y ∈ R}, P lies in second or forth quadrant.
      - If abscissa of a point is zero, the point lies on y axis.
      - If ordinate of a point is zero, the point lies on x axis.
  - - - The coordinates of a point C that divides the line segment joining the points A(x1,y1) and B(x2,y2) in the ratio of ICAI / IABI = k are:
      - x0 = (x1+k.x2) / (1+k) and y0 = (y1+k.y2) / (1+k)
    - - The coordinates of a point C that divides [AB] in the ratio of ICAI / ICBI = k are:
      - x0 = (x1-k.x2) / (1-k) and y0 = (y1-k.y2) / (1-k)
    - - Internal Division:
      - External Division:
  - - - Let A, B, C, D are the verticles of a parallelogram ABCD and K be the intersection of diagonals. Since K is the midpoint of the diagonals:
        
        x1 + x3 = x2 + x4
        
        y1 + y3 = y2 + y4
- - - - The slope of line d is tanO and the slope of line l is tana.
      - DON' FORGET!
        
        For acute angles tanO is positive and for obtuse angles tana is negative.
    - - Vertical Lines are the lines which are parallel to y-axis.
      - l : {(x,y) | x=a , y∈R}
      - m = tan90 = ∞ so that the vertical lines have no slope or we can say the slope of vertical lines are undefined.
      - Horizontal Lines are the lines which are parallel to x-axis.
      - k : {(x,y) | y=b , x∈R}
      - m = tan0 = 0 so that horizontal lines have a slope of 0.
  - - - If d // l m(ABC)=m(DCE). So that, md = tan0, ml = tan0. So m1=m2 then d1 // d2.
      - If d ⊥ l md=tana = IACI / IABI, ml = tan0 = -tanB = -IABI / IACI
    - - If the line is parallel to the x-axis then m = 0 (zero slope)
      - If the line is parallel to the y-axis then m is undefined.
    - - If the points A, B and C are collinear points. then mAB = mAC = mBC
  - - - Let A(x1, y1) be a point on the line l and slope of the line is m, the equation is y - y1 = m . (x - x1)
      - If a point P(a,b) is on the graph of line d, the axis and the ordinate of the point must satisfy the equation of the line.
      - Equations of the lines which are passing through the origins can be found by y = mx (m is gradient of line) formula.
    - - Equation of the line with two points a(x1,y1) and B(x2,y2) on the line is (y-y1) / (y1-y2) = (x-x1) / (x1-x2)
    - - x/a + y/b = 1
    - - ax + by + c = 0 is the standard equation of a line (a, b, and c are constants). This form is usually gotten by manipulating the slope intercept form or the two intercept form.
    - - md is not defined since tan90 = is not defined. x-a=0 since md = (y-b) / (x-b) is not defined x = a.
    - - The line d passing through the point (0,0) has slope m = (y-0) / (x-0) y = mx