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Examples on Angle Between two straight lines.

Illustration:

Draw the lines 3x + 4y – 12 = 0 and 5x + 12y + 13 = 0. Find the equation of the bisector of the angle containing the origin. Also find the acute angle bisector and obtuse angle bisector.

2303_Agnel between two lines.JPG 

Solution:

Let us make the expression on the left-hand side of the given equations of the same sign – or + ve. After substituting x = 0 and 
y = 0.

L.H.S. of (i) is 3.0 + 4.0 – 12 = – 12 = – ve

R.H.S. of (ii) is 5.0 + 12.0 + 13 = 13 = + ve

        So, multiply equation (i) by (–1), we get

                – 3x – 4y + 12 = 0                                              …… (1)

Equation of the bisector of the angle containing origin is given by +ve sign i.e. –3x – 4y+12/5 = + 5x+12y+13/13

 64x + 112y – 91 = 0                                                …… (3)

        Again, the given lines are

                – 3x – 4y + 12 = 0                              …… (1)

                5x –+ 12y + 13 = 0                             …… (2)

To find out whether this is an acute angle bisector or obtuse angle bisector, let us find the sign of a1 a2 + b1 b2 from equation (1) and equation (2).

a1 a2 + b1 b2

= (–3) (5) + (–4) (12) = – 15 – 48 = – 63 = – ve

        the bisector containing the origin is the acute angle bisector.

Now, For obtuse angle bisector, we take –ve origin.

        i.e. –3x – 4y+12/5 = + 5x+12y+13/13

        i.e. 14x – 8y – 221 = 0                                                …… (4)

Well, to confirm all this, let us find angle between one of the lines and one of the bisectors i.e.

5x + 12y + 13 = 0                                              …… (2)

64x + 112y – 91 = 0                                           …… (3)

        Slope of line (2) is m2 = –5/12

        Slope of line (3) is m3 = –64/112

        Let q be the angle between these two lines

                 tan θ = 1402_Equation 1.JPG < 1

         64x + 112y – 91 = 0 is an acute angle bisector.

If θ is the angle between two lines, then tanθ = 1107_Equation 2.JPG   

                             

294_Angle Between two straight lines.JPG

where m1 and m2 are the slopes of the two lines.

        (i)     If the two lines are perpendicular to each other then m1m2 = –1.

Any line perpendicular to ax + by + c = 0 is of the form 
bx – ay + k = 0.

        (ii)    If the two lines are parallel or are coincident, then m1 = m2.

Any line parallel to ax + by + c=0 is of the form ax – ay + k=0.

Let there be two-lines l1 and l2 with slopes m1 and m2 respectively. So tan α = m1, tan β = m2 Angle between them is either

α  β or π – (α  β) depending on the side on considers

419_Two lines of parallel.JPG

Now, tan (a  b) = tan α – tan β/1+tan α tan β

         tan (θ) = m1+m2/1+m1m2                    (α  β θ say)

Since lines can be taken in any order and

tan(– θ) = – tan θ. So only the magnitude of θ can be obtained.

Further tan (π – θ) = – tan θ.

Since magnitude also includes the other angle i.e.

Supplementary angle. So θ is given by

        tan θacute = 887_Equation 3.JPG

Important:

        1.     If lines are parallel

                tan θ = 0  m1 = m2

        2.     If lines are perpendicular

                tan θ = tan (π/2) = ∝

                1 + m1 m2 = 0  m1 m2 = – 1

3.     Equation of a line parallel to y = mx + c is y = mx + k, i.e. Equation of a line parallel to ax + by + c = 0 is ax + by + k = 0

4.     Equation of a line perpendicular to y = mx + c is y = 1/m  x + k i.e. Equation of a line perpendicular to ax + by + c = 0 is 
bx – ay + k = 0

5.     Lines          a1x + b1y + c1 = 0                               …… (i)

                        a2x + b2y + c2 = 0                               …… (ii)

        represents

(i)     intersecting lines if a1/a2 ≠ b1/b2

(ii)    parallel lines if a1/a2 = b1/b2

(iii)    Coincident lines if a1/a2 = b1/b2 = c1/c2

To read more, Buy study materials of Straight Lines comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.

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