HOW TO IF THE PAIR OF STRAIGHT LINE IS PARALLEL OR PERPENDICULAR

Here we are going to see how to check if the pair of straight line is parallel or perpendicular

Two straight lines represented by the equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 are parallel if it satisfies one of the condition given below.

Method 1 :

Angle between a pair of straight lines

θ  =  tan^-1 [2 √(h² - ab)/(a + b)]

If two lines are parallel, then the angle between them will be 0. 

If two lines are parallel, then the angle between them will be 90 degree. 

Question 1 :

Find the combined equation of the straight lines whose separate equations are x − 2y −3 = 0 and x + y+5 = 0.

Solution :

Combined equation of straight lines :

  =  (x − 2y −3)(x + y + 5)

  =  x² + xy + 5x - 2xy -  2y² - 10y - 3x - 3y - 15

  =  x² - xy - 2y² + 2x - 13y - 15

Question 2 :

Show that 4x² + 4xy + y² − 6x − 3y − 4 = 0 represents a pair of parallel lines.

Solution :

4x² + 4xy + y² − 6x − 3y − 4 = 0

By comparing the given equation with the general equation of pair of straight lines 

ax² + 2hxy + by² + 2gx + 2fy + c = 0

We get, a  =  4, b  =  1, 2h  =  4  ==>  h  =  2

θ  =  tan^-1 [2 √(2² - (4)(1))/(4 + 1)]

  =  tan^-1 [2 √(4 - 4)/5]

  =  tan^-1 [0]

  =  0

Hence the given pair of straight line is parallel.

Question 3 :

Show that 2x² + 3xy − 2y² + 3x + y + 1 = 0 represents a pair of perpendicular lines.

Solution :

2x² + 3xy − 2y² + 3x + y + 1 = 0

By comparing the given equation with the general equation of pair of straight lines 

ax² + 2hxy + by² + 2gx + 2fy + c = 0

We get, a  =  2, b  =  -2, 2h  =  3  ==>  h  =  3/2

If two lines are perpendicular then a + b  =  0

2 + (-2)  =  0

Hence the given pair of straight line is perpendicular.

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