HOW TO CHECK IF PAIR OF STRAIGHT LINES INTERSECT

About "How to Check if Pair of Straight Lines Intersect"

How to Check if Pair of Straight Lines Intersect :

Here we are going to see how to check if pair of straight lines intersect.

If the given pair of straight lines ax² + 2hxy + by² + 2gx + 2fy + c = 0 intersect each other, then it must satisfy the condition given below.

abc + 2fgh - af² - bg² - ch²  =  0

Point of intersection :

P(hf − bg/ab − h² , gh − af/ab − h²)

How to Check if Pair of Straight Lines Intersect - Questions

Question 1 :

Show that the equation 2x² −xy−3y² −6x + 19y − 20 = 0 represents a pair of intersecting lines. Show further that the angle between them is tan⁻¹(5).

Solution :

2x² −xy−3y² −6x + 19y − 20 = 0

ax² + 2hxy + by² + 2gx + 2fy + c = 0 

a  =  2, h  =  -1/2, b  =  -3, g  =  -3, f = 19/2, c = -20

Condition for intersection of two lines :

abc + 2fgh - af² - bg² - ch²  =  0

2(-3)(-20)+2(19/2)(-3)(-1/2)-2(19/2)²-(-3)(-3)²-(-20)(-1/2)²  =  0

  =  120 + 57/2 - 2(361/4) + 27 + 20(1/4)

  =  120 + 57/2 - 361/2 + 27 + 5

  =  (240 + 57 - 361 + 54 + 10)/2

  =  (361 - 361) / 2

  =  0

Since it satisfies the above condition, the given pair of straight line is intersected.

Now let us find the angle between them.

 θ =  tan^-1 |2√(h²- ab)/(a + b)|

 θ =  tan^-1 |2√(-1/2)²- 2(-3)/(2 + (-3))|

  =  tan^-1 |2√((1/4) + 6)/(-1)|

  =  tan^-1 |2√(25/4)|

  =  tan^-1 |(2(5)/2)|

  =  tan^-1 5

Hence proved.

Question 2 :

Prove that the equation to the straight lines through the origin, each of which makes an angle α with the straight line y = x is x² − 2xy sec 2α + y² = 0

Solution :

Equation of the line AB :

y  =  m₁ x

m₁ x - y  =  0

Equation of the line CD :

y  =  m₂ x

m₁₂ x - y  =  0

In order to find the slope of first line, let us find the angle between x-axis and the line AB.

θ  =  45 - α   

m₁  =  tan θ  =  tan (45 - α)  ---(1)

Angle between the line CD and x-axis

θ  =  45 + α   

m₂  =  tan θ  =  tan (45 + α)  ---(2)

tan (45 - α)  =  (tan 45 - tan α)/(1 + tan 45 tan α)

m₁  =  (1 - tan α)/(1 + tan α)

tan (45 + α)  =  (tan 45 + tan α)/(1 - tan 45 tan α)

m₂  =  (1 + tan α)/(1 - tan α)

Equation of straight lines :

(m₁ x - y) (m₂ x - y)  =  0

m₁m₂ x²-m₁xy - m₂xy  + y² =  0

m₁m₂ x² - xy(m₁ + m₂) + y² =  0   ----(3)

m₁ + m₂  =  (1 - tan α)/(1 + tan α) + (1 + tan α)/(1 - tan α)

  =  ((1 - tan α)² + (1 + tan α)²)/(1 + tan α)(1 - tan α)

  =  (1 - 2tan α + tan ²α + 1 + 2tan α + tan ²α)/(1-tan²α)

  =  (2 + 2tan ²α)/(1-tan²α)

  =  2[(1+tan²α)/(1-tan²α)]

  =  2 sec 2α

Note :

cos 2α  =  (1-tan²α) / (1+tan²α)

m₁ m₂  =  ((1 - tan α)/(1 + tan α)) ((1 + tan α)/(1 - tan α))

  =  1

By applying the above values in equation 3, we get the required equation.

(1) x² - xy 2 sec 2α + y² =  0

After having gone through the stuff given above, we hope that the students would have understood "How to Check if Pair of Straight Lines Intersect". 

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