MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
Menu
Get extra Rs. 148 off
USE CODE: SELF10

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.

Acute angle bisector and Obtuse angle bisector

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   

Angle between two lineswhere 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

Two lines are parallel

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.

Get Extra Rs. 1,590 off

COUPON CODE: SELF10


Course Features

  • 731 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution