Straight Lines, Objectives, Strainght Line Parallel to an Axis, Derivation of the Equation of Straight Line in Various Standard Forms

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Objectives

  • After studying this lesson, you will be able to:

  • Derive equations of a line parallel to either of the coordinate axes;

  • Derive equations of a line in different forms (slope-intercept, point-slope, two point, intercept, and perpendicular)

  • Find the equation of a line in the above forms under given conditions;

  • State that the general equation of first degree represents a line;

  • Express the general equation of a line into

    • Slope-intercept form

    • Intercept form and

    • Perpendicular form;

  • Derive an expression for finding the distance of a given point from a given line;

  • Calculate the distance of a given point from a given line;

  • Derive the equation of a line passing through a given point and parallel/perpendicular to a given line;

  • Find equation of family of lines passing through the point of intersection of two lines.

Strainght Line Parallel to an Axis

  • If you stand in a room with your arms stretched, we can have a line drawn on the floor parallel to one side. Another line perpendicular to this line can be drawn intersecting the first line between your legs.

  • In this situation the part of the line in front of you and going behind you is the y-axis and the one being parallel to your arms is the x-axis.

  • The direction part of the y-axis in front of you is positive and behind you is negative.

  • The direction of the part x-axis to your right is positive and to that to your left is negative.

Now, let the side facing you be at units away from you, then the equation of this edge will be (parallel to x-axis) where is equal in absolute value to the distance from the x-axis to the opposite side.

  • If , then the line lies in front of you, i.e., above the x-axis.

  • If , then the line lies behind you, i.e., below the x-axis.

  • If , then the line passes through you and is the x-axis itself.

Again, let the side of the right of you is at units apart from you, then the equation of this line will be (parallel to y - axis) where is equal in absolute value, to the distance from the y-axis on your right.

  • If , then the line lies on the right of you, i.e., to the right of y-axis.

  • If , then the line lies on the left of you, i.e., to the left of y-axis

  • If , then the line passes through you and is the y-axis.

Example:

Find the equation of the line passing through and (i) parallel to x-axis (ii) parallel to y-axis

Solution:

(i) The equation of any line parallel to x-axis is

Since it passes through, hence b

The required equation of the line is

(ii) The equation of any line parallel to y-axis is

Since it passes through, hence

The required equation of the line is

Derivation of the Equation of Straight Line in Various Standard Forms

  • So far we have studied about the inclination, slope of a line and the lines parallel to the axes. Now the questions is, can we find a relationship between and , where is any arbitrary point on the line?

  • The relationship between x and y which is satisfied by the co-ordinates of arbitrary point on the line is called the equation of a straight line. The equation of the line can be found in various forms under the given conditions, such as

    • When we are given the slope of the line and its intercept on y-axis.

    • When we are given the slope of the line and it passes through a given point.

    • When the line passes through two given points.

    • When we are given the intercepts on the axes by the line.

    • When we are given the length of perpendicular from origin on the line and the angle which the perpendicular makes with the positive direction of x-axis.

  • We will discuss all the above cases one by one and try to find the equation of line in its standard forms.

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