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Mathematics Dictionary
Dr. K. G. Shih
Point in Mathematics
Subjects

Answers


Q01. One point (x,y) in rectangular coordinates
  • P(x,y) is a point in rectangular coordinates.
  • x is called abissica and y is called ordinate.
  • Quadrant and (x,y).
    • 1st quadrant : top of x-axis and right side of y-axis.
    • 2nd quadrant : top of x-axis and left side of y-axis.
    • 3rd quadrant : bottom of x-axis and left side of y-axis.
    • 4th quadrant : bottom of x-axis and right side of y-axis.
  • Quadrant and (x,y).
    • 1st quadrant : x = + and y = +
    • 2nd quadrant : x = - and y = +
    • 3rd quadrant : x = - and y = -
    • 4th quadrant : x = + and y = -
Exercise : Draw point P(-2,-3) in rectangular coordinates.
 Go to Begin

Q02. One point (R,A) in polar coordinates
Construction
  • Draw Ox and Oy coordinate.
  • Draw point P. Join PO and let PO = R.
  • Let angle POx = A
  • Coordinate (R,A) is the point P.
Realtion with rectangular coordinate
  • x = R*cos(A) and y = R*sin(A).
  • R = Sqr(x^2 + y^2).
  • A = arctan(y/x).
Exercise : Use calculator to draw R = sin(A) with A = 0,30,45,60,90, 120,135,150,180 in degrees Go to Begin

Q03. One point z = a + b*i in complex number coordinates
Construction
  • Draw real number axis Ox and imaginary number axis Oy.
  • Draw point P. Join PO and let PO = R.
  • Let angle POx = A
  • Point P is given z = a + b*i where i is Sqr(-1).
Relation with polar form
  • a = R*cos(A) and b = R*sin(A).
  • R = Sqr(a^2 + b^2).
  • A = arctan(b/a).
  • Hence z = R*(cos(A) + i*sin(A)) = R*cis(A).
Exercises
  • Draw points z1 = 2 + i and z2 = 2 - i.
  • Convert z = 1 - i to R*cis(A).
Study subjects
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Q04. One point x = F(t) and y = G(t) in parametric system
  • The x and y is same as rectangular coordinates.
  • F(t) is a function and G(t) is also a function.
Examples
  • Study Find the name of the curve of y = sec(t) and y = tan(t).
Go to Begin

Q05. Two points (x1,y1) and (x2,y2)
Distance between two points
  • D = Sqr((x2-x1)^2 + (y2-y1)^2)
Slope between two points
  • Slope = (y2-y1)/(x2-x1) where x1 is not equal to x2.
  • If x1 = x2 the two points will make vertical line.
  • if y1 = y2 the two point will make horizontal line
  • if slope is greater than zero, the line is defined as increasing.
  • if slope is less than zero, the line is defined as decreasing.
One fixed point and one moving point. Go to Begin

Q06. Three points and locus of line
  • Two given points A and B.
  • A point P has same distance from A and B.
  • What is the locus of P ?
  • It the bisector of line AB.
Go to Begin

Q07. Three points define a circle

Geometrical construction
 Solve three linear equations to get equation of circle.
  • Let equation of circle be x^2 + y^2 + a*x + b*y + c = 0.
  • Substitute three points into above equation.
  • We get 3 linear equation about a, b, c.
  • Solve the linear equations we get a, b, c.
  • Use completing the square to find center and radius.
  • Study How to find equation of circle ?
Go to Begin

Q08. Three points define an ellipse
Go to Begin

Q09. Three points define a hyperbola
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Q10. Three points define a locus of arc

 Go to Begin

Q11. Three points define a locus of arc

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Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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