Calculus With Analytical Geometry | Existence the Pair of Lines | How to Find Pair of Lines? | Angle Between the Straight Lines | Exercise 6.1 | Questions 1 to 8

1. Calculus With Analytical Geometry
PLANE CURVE-I
PAIR OF LINES & ANGLE BETWEEN THE STRAIGHT LINES
Existence the Pair of Lines:
The most general equation of second degree in is:
( )
We have the required eliminant:
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This is the condition that represents a pair of straight lines.
How to find the pair of Lines?
Suppose and re-write the equation ( ) as a quadratic equation in such that:
( )
Or ( ) ( )
Compare the above equation with
So ( )
By Quadratic Formula
√
( ) √[ ( )] ( )
This equation represents the pair of lines.

2. Examine whether each of the given equations represents two straight lines. If so,
find an equation of each straight line ( Problems 1-5 )
Q:
Sol:
It can be also written as ( ) ( ) ( ) ( )
Compare it with ( )
By Comparing ( ) ( ), we have
We have the required eliminant:
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( ) ( ) ( )
Thus there exists a pair of lines.
Equation ( )
( )
Compare it with
So ( )
By Quadratic Formula
[ ( )] √[ ( )] ( )( )
( )
( ) √
( ) √

3. ( ) √( ) ( )( ) ( )
( ) √( )
( ) ( )
Either
( ) ( )
or
( ) ( )
Required pair of lines
For what value of will each of the following equations represent a pair of
straight lines ( Problems 6-8 )
Q: ( )
Sol: ( )
It can also be written as ( ) ( ) ( )
Compare it with ( )
By Comparing ( ) ( ), we have
As the given equation represent a pair of straight line
So | |

4. | |
By Laplace Expansion ( ) ( ) ( )
or
or
or
Angle between the straight lines:
The second degree in is:
( )
Let be an angle between the two straight lines represented by equation ( ) is given by
√
Note:
 The Two straight lines are at right angles if
 The Two straight lines are parallel if
Find the angle between each of the following pairs of lines ( Problems 9-13 )
Q:
Sol:
It can also be written as ( ) ( ) ( )
Compare it with ( )
By Comparing ( ) ( ), we have
( )
As ( ) ,
So the two straight lines are at right angle ( )

5. Q:
Sol:
It can also be written as ( ) ( ) ( ) ( ) ( )
Compare it with ( )
By Comparing ( ) ( ), we have
Now
√ √( ) ( )( ) ( )
( )