Straight Lines, Equation of a Straight Line Parallel to the Given Line, Equation of a Straight Line Perpendicular to the Given Line

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Equation of a Straight Line Parallel to the Given Line

Let

Be any line parallel to the given line,

The condition for parallelism of (1) and (2)

(Say)

With these values of and 1, (1) gives

or

Or where

This is a line parallel to the given line. From equations (2) and (3) we observe that

(i) Coefficients of and are same

(ii) Constants are different, and are to evaluate from given conditions.

Example:

Find equation of the straight line, which passes through the point and which is parallel to the straight line .

Solution:

Equation of any straight line parallel to the given equation can be written if we put

(i) The coefficients of and as same as in the given equation.

(ii) Constant to be different from the given equation, which is to be evaluated under given condition.

Thus, the required equation of the line will be, for some constant

Since it passes through the point hence, or

Required equation of the line is .

Equation of a Straight Line Perpendicular to the Given Line

Let

Be any line parallel to the given line,

The condition for parallelism of (1) and (2)

(Say)

With these values of and , (1) gives

Or where

Hence, the line (3) is perpendicular to the given line (2)

We observe that in order to get a line perpendicular to the given line we have to follow the following procedure:

(i) Interchange the coefficients of and

(ii) Change the sign of one of them.

(iii) Change the Constant term to a new constant (say), and evaluate it from given condition.

Example:

Find the equation of the line which passes through the point and is perpendicular to the line 0.

Solution:

Following the procedure given above, we get the equation of line perpendicular to the given equation as

(i) Passes through the point , hence

0 or

Required equation of the straight line is .

Equation of Family of Lines Passing through the Point of Intersection of Two Lines

Let

And , be two intersecting lines.

Let be the point of intersection of and , then

And

Now consider the equation

The point of intersection of two lines

The Point of Intersection of Two Lines

The point of intersection of two lines

It is a first degree equation in and . So it will represent different lines for different values of . If we replace by and by we get

using (3) and (4) in (6) we get

which is true.

So equation (5) represents a family of lines passing through the point i.e. the point of intersection of the given lines and .

A particular member of the family is obtained by giving a particular value to . This value of can be obtained from other given conditions.

Example:

Find the equation of the line passing through the point of intersection of the lines and and containing the point .

Solution:

Equation of family of lines passing through the intersection of given lines is

This line will contain the point if

i.e.

Therefore the equation of required line is,

i.e.

i.e.

Or

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