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The Straight Line Contd...

Angle between Two Straight Lines:

 

(1) If a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 are the equations of two given lines,

      the angle (theta) between the lines is given by

 

     cos (theta) = [Modulus of ((a1*a2) + (b1*b2))] / [sqrt((a1^2+b1^)(a2^2 +b2^2))]

 

(2) If m1 and m2 are the slopes of two lines, the angle (theta) is

     given by tan (theta) = (m1-m2)/(1+m1*m2)

 

(3) Condition for parallel lines is (a1/a2) = (b1/b2) or m1 = m2

 

(4) Condition for Perpendicular lines is (a1*a2 + b1*b2) = 0 or m1*m2 = -1

 

(5) The equation of a line through the point (x1, y1)  & parallel to the line

      ax+by+c =0 is given by a(x-x1) + b(y-y1) = 0

 

(6) The equation of a line through the point (x1, y1) & perpendicular to the line

      ax+b+c = 0 is b(x-x1)-a(y-y1) = 0

 

 Solved Examples:

 

(1) Show that the lines x+2y+3 = 0 and 2x+4y+6 = 0 are parallel

 

Solution:

 

Here a1=1, b1=2 & a2=2, b2=4

            (a1/a2)= ½ & (b1/b2) = 2/4=1/2

Condition for parallelism is (a1/a2)=(b1/b2), which is equal =1/2, hence we can say that the above two lines are parallel.

 

(2) Show that the lines 3x+y-4 = 0 & x-3y+1 = 0 are perpendicular

 

Solution:

 

Here a1 = 3, b1 = 1 & a2 = 1, b2 = -3

The condition for perpendicular lines is (a1*a2+b1*b2) = 0

Therefore (3*1) + (1*(-3)) = 3-3 = 0, therefore we can say that the given lines are perpendicular.

 

(3) Find the equation of the line through the point (-1, 2) and parallel to 2x-3y+1 = 0

 

Solution:

 

Here a = 2, b = -3 & x1 = -1, y1 = 2

The required equation is a(x-x1) + b(y-y1) = 0

Substituting the values on the above we get 2(x+1) -3(y-2) = 0

On simplification we get the equation as 2x-3y+8 = 0

 

(4) Find the equation of the line through the point (1, 2) and perpendicular to the line

      3x+2y-1 = 0

 

Solution:

 

Here a =3, b =2 & x1 =1, y1 =2

The required equation is b(x-x1)-a(y-y1) = 0

Substituting the values on the above we get 2(x-1)-3(y-2) = 0

On simplification we get the equation as 2x-3y+4 =0.

 

navin
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navin said:

aby saale sabse aasan paper h yaar

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