Note on Equation of a Straight Line

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Straight Line

Straight line
Straight line

The equation of the straight line can be calculated in different ways according to the given condition. The different conditions of equation of straight line are given below:

Horizontal, Vertical and Oblique lines

  1. The equation of the straight line parallel to X-axis: Equation of the straight line parallel to X-axis and passing through the point (o, b) cutting at Y-axis i.e. at y = b.
  2. The equation of the straight line parallel to Y-axis: Equation of the straight line parallel to Y-axis and passing through the point (a, O) cutting at X-axis is x=a.
  3. The oblique line: A line which is neither parallel to x-axis nor parallel to they-axis is called an oblique line.

Inclination of a line

Inclination of a plane

When the angle is made in a straight line with x-axis making positive direction is said to be the inclination of a line. It is devoted by θ.

Slope of Straight line

Slope of a straight line
Slope of a straight line

The tangent of the angle made by the straight line on the positive X-axis is called slope of the straight line. It is denoted by m. If θ be the angle made by the straight line AB on X-axis, then slope AB = m = tanθ

Collinear Point

Collinear Point
Collinear Point

If three or more than three points lies in same straight line then it is said to be collinear points. We can prove the point P, Q, and R collinear by using slope as a slope of PQ = slope of QR =` slope of PR. As the points P, Q, and R in collinear. If the points P, Q, and R lie on the same line the slope of the two line will equal.

Intercepts made by a line on the axis

Intercepts made by a line on the axis
Intercepts made by a line on the axis

If the line AB cuts the X and Y - axis at the points A (a, O) and B(0, b) respectively. Then the length from the origin to the point of intersection of the line AB and X - axis is called the x -intercept and length from the point of intersection of the line AB and Y - axis are called y - intercept. In the figure , x -intercept OA = a and y -intercept OB = b.

Convention for the signs of intercepts

  • If the X-intercepts is measured in right side from the origin then it is considered as positive and measured on the left side from the origin then it is considered as negative.
  • If the Y-intercept is measured over the origin then it is considered as positive and measured on the under the origin then it is considered as negative.

 

Equation of a straight line parallel to X-axis

Equation of a straight line parallel to x-axis
Equation of a straight line parallel to x-axis

Let AB be a straight line parallel to X-axis. Then the ordinate of every point on the line XY is constant say b.
Let P(x, y) be any point on the lie AB. From P, draw PM perpendicular to X-axis, then MP = y.
∴ y = b, which is required equation of the line AB. The line AB. The line y = b lies above or below the X-axis according to as b is positive or negative.
If b = 0, then the line Ab coincides with X-axis. So, the equation of X-axis is y = 0.


Equation of straight line parallel to Y-axis

Equation of a straight line parallel to y-axis
Equation of a straight line parallel to y-axis

Let AB be a straight line parallel to Y-axis. Then, the abscissa of every point on the line AB is constant, say a.
Let P(x, y) be any point on the AB. From P, draw PN perpendicular to Y-axis, then NP = x.
∴ x = a, which is required equation of the line AB. The line x = a lies to the right or left of Y-axis according to as a is positive or negative.
If a = 0, then the line Ab coincides with y-axis.

The equation of the straight line in the standard form: There are three standard forms of the equation of the straight line.They are:
Slope - intercept form: If the slope of the straight line m = tanθ and y - intercept (c) are known, the equation of the straight line in slope-intercept form is y = mx + c.

  1. If the straight line passes through the origin then y - intercept of c = 0. In this case, an equation of the straight line is y = mx.
  2. The tangent of the angle made by the straight line on the positive X-axis is called slope of the straight line. 
  3. When the angle is made in a straight line with x-axis making positive direction is said to be the inclination of a line. It is devoted by θ.
  4. If three or more than three points lies in same straight line then it is said to be collinear points.
.

Very Short Questions

Here, given points (2,5)

The equation of the straight line parallel to y-axis is x=x coordinatesof given points

or, x=2

∴x-2=0Ans.

Here,

given points (5,-2)

The equation of the straight line parallel to y-axis is x=x coordinatesof given points

or, x=5

∴x-5=0Ans.

Given points,(-3,2)

The equation of the straight line parallel to x-axis is y=y-coordinates of the point.

or, y=2

∴ y-2=0.Ans.

Given points,(-3,2)

The equation of the straight line parallel to x-axis is y=y-coordinates of the point.

or, y=-4

∴ y+4=0.Ans.

Here,

The equation of the straight line parallel to x-axis and 4 units above the origin is y=4 and 4 units below the origin is y=-4.

∴The required equation are y-4=0 and y+4=0.Ans.

Here,

Angle on x-axis (θ)=45° and slope of the line(m)=tan 45θ°=1

Equation of straight line passing through origin is y=mx.

or, y=1,x

∴y-x=0.Ans.

Angle on x-axis (θ)=150°

∴ Slope (m)=tanθ=tan150°=-\(\frac{1}{√3}\)

Using formula,y=mx

y=-\(\frac{1}{√3}\)x

or,√3y=-x

∴x+√3y=0.Ans.

Here,

Given equation,x-y=5

or, -y=-x+5...........(i)

Comparing equation (i) with y = mx+c, we get

m=1 and c=-5

∴Slope (m)=1 and y-intercept (c)=-5.Ans.

Here given equation, y=√3 x or y=√3 x+0........(i)

Comparing equation (i) with y=mx +c,we get,

Slope (m)=√3 and y-intercept(c)=0 Ans.

0%
  • The line AB whose point are A(0,1) and B(1,3) is produced to D(10,k).Find the value of k.

    21


    12


    8


    3


  • Find the value of x if the slope of the line joining the point A(1,3) and B(x,6) is 1.

    2


    9


    10


    4


  • The slope of the line joining A(-2,4) and B(3,5) is equal to the slope of the line joining C(0,4) and D(-3,k). Find the value of k.

    12/5


    17/5


    11/5


    17/6


  • Given points P(3,2) Q(0,-4) and R(-3,x) are collinear.Find the value of x.

    x=-19


    x=-18


    x=-10


    x=10


  • Given points (-5,1),(5,5) and (k,7) are collinear.Find the value of k.

    k=1


    k=-9


    k=10


    k=2


  • Find the equation of straight line having slope -3 and passing through (2,-2).

    3x-y=9


    5x+y=4


    3x-y=2


    3x+y=4


  • The straight line y =mx+6 passes through the points (1,4) and (-2,-5).Determine the equation of the straight line.

    y=3x+7


    y=2x+1


    y=3x+1


    y=3x-1


  • The straight line y =mx+6 passes through the points (3,4) and (-2,6).Determine the equation of the straight line.

    2x+4y=26


    2x-5y=12


    2x-9y=21


    2x+5y=26


  • Find the equation of a straight line making an angle 30° with X-axis and passing through the mid-point of the line joining (-2,3) and (8,5).

    x√3-y=3+4√3


    x√3-y=2-2√3


    x√2-y=3-4√2


    x√3-y=3-4√3


  • Find the equation of a straight line making an angle of  45° with X-axis and passing through the mid-point of the line joining (-2,3) and (4,1).

    x-y+2=0
    x+y+1=0
    x-y+4=0
    x-y-1=0
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Aiyaan

Find the equation of the line joining the origin and the points of trisection of join of (1,4) and (2,3).


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Popular Chaube

How to find out the equation of a straight line having equation and a point on X axis?


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