How to Check 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 ax^{2} + 2hxy + by^{2} + 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 √(h^{2} - 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)
= x^{2} + xy + 5x - 2xy - 2y^{2} - 10y - 3x - 3y - 15
= x^{2} - xy - 2y^{2} + 2x - 13y - 15
Question 2 :
Show that 4x^{2} + 4xy + y^{2} − 6x − 3y − 4 = 0 represents a pair of parallel lines.
Solution :
4x^{2} + 4xy + y^{2} − 6x − 3y − 4 = 0
By comparing the given equation with the general equation of pair of straight lines
ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0
We get, a = 4, b = 1, 2h = 4 ==> h = 2
θ = tan^{-1} [2 √(2^{2} - (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 2x^{2} + 3xy − 2y^{2} + 3x + y + 1 = 0 represents a pair of perpendicular lines.
Solution :
2x^{2} + 3xy − 2y^{2} + 3x + y + 1 = 0
By comparing the given equation with the general equation of pair of straight lines
ax^{2} + 2hxy + by^{2} + 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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