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Straight Line

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THE STRAIGHT LINE

A straight line is the simplest geometric curve.

FORMS OF A STRAIGHT LINE

  1. A STRAIGHT LINE PARALLEL TO THE AXES AT A GIVEN DISTANCE

    2. The equation of a straight line parallel to the x-axis and at a distance a from it is

     

    y = a

    NOTE: By putting b = 0, we can find the equation of the x-axis as y = 0.

    Similarly for a straight line parallel to the y-axis and at a distance b from it the equation is

     

    x = a

    By putting b = 0, we can find the equation of the y-axis as x = 0.

  2. A STRAIGHT LINE PASSING THROUGH THE ORIGIN

The equation of a straight line passing through the origin is given by

 

y = mx

SLOPE

If a straight line AB makes an angle q with the positive direction of the x-axis then the slope of the line is given by tan q and is denoted by the letter m.

The slope of a line passing through the two points (x1,y1), (x2,y2) is given by

 

tan q = (y2-y1)/(x2-x1)

Thus, m = (y2-y1)/(x2-x1)

NOTE:

  1. if two lines are parallel, their slopes are equal ( because the lines are equally inclined to the positive direction of the x-axis)

  2. if two lines are perpendicular the product of their slopes is -1

 

SPECIAL FORMS OF A STRAIGHT LINE

  1. THE SLOPE-INTERCEPT FORM

    2. The equation of a straight line whose gradient is m and whose intercept on the y-axis is c is given by

     

    y = mx + c

  2. POINT SLOPE FORM

    3. The equation of a straight line passing through the given point (x1,y1) and having a slope m is given by

     

    y-y1 = m(x- x1)

     

  3. THE TWO POINT FORM OF THE EQUATION OF A STRAIGHT LINE

    4. The equation of a line passing through the points (x1,y1), (x2,y2) is given by

     

    (y-y1) / (x - x1) = (y1 - y2)/(x1 - x2)

  4. PERPENDICULAR OR THE NORMAL FORM

    If p is the length of the perpendicular from the origin upon a straight line, and a is the angle the perpendicular makes with the axis of x, then the equation of the straight line is given by

     

    Xcosa + ysina = p

  5. THE POINT ANGLE FORM

    6. If (x1-y1) is a given point on a straight line, (x,y) be any general points on the line ,q is the inclination of the line and r the distance between the points (x,y), (x1,y1) then the equation of the line is given as

     

    (x-x1)/(cos q) = (y-y1)/(sinq) = r

  6. THE INTERCEPT FORM

The equation of a straight line which makes known intercepts on the axes of coordinates is given by the equation

( If the line make intercepts a and b on the axes of x and y respectively. Let P(x,y) represent any point on the line then, )

 

X/a + y/b = 1

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