**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 ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 intersect each other, then it must satisfy the condition given below.

abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0

Point of intersection :

P(hf − bg/ab − h^{2} , gh − af/ab − h^{2})

**Question 1 :**

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

**Solution :**

2x^{2} −xy−3y^{2} −6x + 19y − 20 = 0

ax^{2} + 2hxy + by^{2} + 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 - af^{2} - bg^{2} - ch^{2} = 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√(h^{2}- 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 x^{2} − 2xy sec 2α + y^{2} = 0

**Solution :**

Equation of the line AB :

y = m_{1} x

m_{1} x - y = 0

Equation of the line CD :

y = m_{2} x

m_{12} 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

m_{1 } = tan θ = tan (45 - α) ---(1)

Angle between the line CD and x-axis

θ = 45 + α

m_{2 } = tan θ = tan (45 + α) ---(2)

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

m_{1} = (1 - tan α)/(1 + tan α)

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

m_{2} = (1 + tan α)/(1 - tan α)

Equation of straight lines :

(m_{1} x - y) (m_{2} x - y) = 0

m_{1}m_{2 }x^{2}-m_{1}xy - m_{2}xy + y^{2} = 0

m_{1}m_{2 }x^{2 }- xy(m_{1 }+ m_{2}) + y^{2} = 0 ----(3)

m_{1 }+ m_{2 }= (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 ^{2}α + 1 + 2tan α + tan ^{2}α)/(1-tan^{2}α)

= (2 + 2tan ^{2}α)/(1-tan^{2}α)

= 2[(1+tan^{2}α)/(1-tan^{2}α)]

= 2 sec 2α

Note :

cos 2α = (1-tan^{2}α) / (1+tan^{2}α)

m_{1 }m_{2 }= ((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)_{ }x^{2 }- xy 2 sec 2α + y^{2} = 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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