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Session Objectives

- Equations of bisectors of angles between two lines
- Acute/obtuse angle bisectors
- Position of origin w.r.t bisecors
- Equation of family of lines through intersection of two lines
- Pair of lines - locus definition
- Pair of lines represented by second degree equation
- Angle between two lines, represented as a second degree equation

a2x+b2y+c2=0

N

A

P(h,k)

a1x+b1y+c1=0

M

X’

O

X

Y’

Equation of BisectorConsider two lines a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0

We are required to find the equations of the bisectors of the angle between them.

The required equations are the equations to the locus of a point P(h, k)

equidistant from the given lines.

Acute/obtuse Angle Bisectors

Algorithm to determine equations of bisectors of acute angle and obtuse angle between a pair of lines.

Step I : Rewrite the equations of the lines in general form a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 such that c1 and c2 are positive.

Step II : Determine sign of expression a1a2+b1b2

Step III : Write the equations of the bisectors

;

Acute/obtuse Angle Bisectors

Step IV : Case (i) a1a2+b1b2 > 0

Obtuse angle bisector

Acute angle bisector

Acute/obtuse Angle Bisectors

Step IV : Case (i) a1a2+b1b2 < 0

Acute angle bisector

Obtuse angle bisector

Illustrative Example

Find the equation of the obtuse angle bisector of lines 12x-5y+7 = 0 and 3y-4x-1 = 0.

Solution :

Rewrite the equations to make

the constant terms positive,

12x-5y+7 = 0 and

4x-3y+1 = 0

Calculate a1a2+b1b2

12*4+(-5)*(-3) = 63

Solution Cont.

a1a2+b1b2 > 0, therefore the obtuse angle bisector is

Simplifying,

Which is the required equation of the obtuse angle bisector.

Origin w.r.t. Angle Bisectors

Algorithm to determine whether origin lies in the obtuse angle or the acute angle between a pair of lines

Step I : Rewrite the equations of the lines in general form a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 such that c1 and c2 are positive.

Step II : Determine sign of expression a1a2+b1b2

Origin w.r.t. Angle Bisectors

Step III : Case (i) a1a2+b1b2 > 0

Origin lies in obtuse angle between the lines

Origin w.r.t. Angle Bisectors

Step III : Case (i) a1a2+b1b2 < 0

Origin lies in acute angle between the lines

Illustrative Example

For the straight lines 4x+3y-6 = 0 and 5x+12y+9 = 0 find the equation of the bisector of the angle which contains the origin.

Solution :

Rewrite the equations to make

the constant terms positive,

-4x-3y+6 = 0 and

5x+12y+9 = 0

Calculate a1a2+b1b2

(-4)*5+(-3)*(12) = -56

a1a2+b1b2 < 0, therefore origin lies in the acute angle.

Illustrative Example

Acute angle bisector is given by :

The origin lies in the acute angle and the equation of the acute angle bisector is 7x+9y-3 = 0.

Family of Lines Through Intersection of a Pair of Lines

Equation of the family of lines passing through the intersection of the lines a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 is given by

can be calculated using some given condition

Illustrative Example

Find the equation of the straight line which passes through the point (2, -3) and the point of intersection of x+y+4 = 0 and 3x-y-8 = 0.

Solution :

Required equation can be written as x+y+4+(3x-y-8) = 0 where is a parameter.

This passes through (2, -3).

2-3+4+ (3*2+3-8) = 0.

= -3

the required equation is x+y+4-3(3x-y-8) = 0

or –8x+4y+28 = 0 or 2x-y-7 = 0

Pair of Lines - Locus Definition

A pair of straight lines is the locus of a point whose coordinates satisfy a second degree equation ax2+2hxy+by2+2gx+2fy+c = 0 such that it can be factorized into two linear equations.

Pair of Lines

ax2+2hxy+by2+2gx+2fy+c = 0 in general represents all the conics in the x-y plane.

Individual Lines

To find the equation of individual lines in the pair of lines

ax2+2hxy+by2+2gx+2fy+c = 0,

Method I :

Step I : Rewrite the equation as a quadratic in x (or y).

Step II : Solve for x (or y).

Method II :

Will be discussed later.

Illustrative Example

Find the separate equations of the lines represented by 2x2-xy-y2+9x-3y+10 = 0.

Point of Intersection

Point of intersection of the pair of lines represented by

ax2+2hxy+by2+2gx+2fy+c = 0 is

No need to memories this formula. To find the required point, find the equations of the individual lines and solve simultaneously

Homogeneous Equation

An equation, with R.H.S. 0, in which the sum of the powers of x and y in every term is the same, say n, is called a homogeneous equation of nth degree in x and y.

Pair of Lines Through Origin

A pair of straight lines passing through the origin is represented by a homogeneous equation of second degree

ax2+2hxy+by2 = 0

ax2+2hxy+by2 = 0 can be rewritten as b(y-m1x)(y-m2x) = 0, where m1 and m2 are the slopes of the two lines.

bm1m2x2-b(m1+m2)xy+by2 = 0

The above relations can be used to find the equations and the slopes of the individual lines.

Illustrative Example

Find the separate equations of the lines represented by 4x2+24xy+11y2 = 0.

Solution :

Pair of Lines Through Origin

If a pair of straight lines is represented by

ax2+2hxy+by2+2gx+2fy+c = 0, then

ax2+2hxy+by2 = 0

represents a pair of lines parallel to them and passing through the origin.

Individual Lines

To find the equation of individual lines in the pair of lines

ax2+2hxy+by2+2gx+2fy+c = 0,

Method II :

Step I : Find slopes of the individual lines m1 and m2 using ax2+2hxy+by2 = 0

Step II : compare coefficients in the identity ax2+2hxy+by2+2gx+2fy+c b(y-m1x-c1)(y-m2x-c2) to find c1 and c2.

Illustrative Example

Find the separate equations of the lines represented by 2x2+5xy+3y2+6x+7y+4 = 0.

Solution :

Solution Cont.

Consider the identity

X’

O

X

Y’

Pair of Lines With Given ConditionTo find the joint equation of a pair of lines joining the origin to the points of intersection of the curve ax2+2hxy+by2+2gx+2fy+c = 0

and the line lx+my+n = 0.

X’

O

X

Y’

Pair of Lines With Given ConditionStep I : Rewrite lx+my+n = 0 such that R.H.S. = 1

Step II : Make the equation of the curve homogeneous

X’

O

X

Y’

Pair of Lines With Given ConditionThe required equation is the equation arrived at in Step II.

Illustrative Example

Find the equation of the lines joining the origin to the points of intersection of the straight line y = 3x+2 and the curver x2+2xy+3y2+4x+8y-11 = 0.

Solution :

Equation of straight line can be rewritten as

Using this to make the equation of the curve homogeneous,

Angle Between a Pair of Lines

If ax2+2hxy+by2+2gx+2fy+c = 0 represents a pair of lines,

Acute angle between the lines is given by :

Independent of g, f, c

Angle Between a Pair of Lines

If ax2+2hxy+by2+2gx+2fy+c = 0 represents a pair of lines,

Acute angle between the lines is given by :

Lines are parallel or coincident if h2 = ab

Angle Between a Pair of Lines

If ax2+2hxy+by2+2gx+2fy+c = 0 represents a pair of lines,

Acute angle between the lines is given by :

Lines are perpendicular if a+b = 0

Acute angle between the pair of lines is given by

Illustrative ExampleFind the angle between the pair of lines represented by 2x2+5xy+3y2+6x+7y+4 = 0.

Solution :

Angle Bisectors of a Pair of Lines

If ax2+2hxy+by2 = 0 represents a pair of lines,

Equation of angle bisectors is given by :

Perpendiculars to a Pair of Lines

If ax2+2hxy+by2 = 0 represents a pair of lines,

Equation of perpendiculars to the pair of lines is given by :

Illustrative Example

Find the equation of the bisectors of the lines represented by 135x2-136xy+33y2 = 0.

Solution :

Equation of bisectors are given by :

Which is the required equation.

Class Exercise - 1

The bisectors of the angle between the lines y = 3x+3 and 3y = x+33 meet the X-axis at P and Q. Find length PQ.

Solution :

The equations can be rewritten as

Angle bisectors are given by

Class Exercise - 2Prove that the bisectors of the angles formed by any two intersecting lines are perpendicular to each other.

Solution :

A. Consider two intersecting lines

Consider the angle bisectors

+++ =

+ = /2

Q.E.D.

Class Exercise - 3

Show that the reflection of the lines px+qy+r = 0 in the line x+y+1 = 0 is the line qx+py+(p+q-r) = 0, where p -q

Solution :

Angle bisectors of the lines px+qy+r = 0 and qx+py+(p+q-r) = 0 are

Second equation can be written as x+y+1 = 0 (as p+q 0)

Q.E.D.

Class Exercise - 4

A rhombus has two of its sides parallel to the lines y = 2x+3 and y = 7x+2. If the diagonals cut at (1,2) and one vertex is on the Y-axis, find the possible values of the ordinate of that vertex.

Solution :

Concept : Diagonals of a rhombus bisect the angle.

Let the sides intersecting at the Y-axis be 2x-y+1 = 0 and 7x-y+2 = 0.

These lines will meet the Y-axis at (0,1) and (0,2).

1 = 2 = (say)

Thus the sides are 2x-y+ = 0 and 7x-y+ = 0.

Class Exercise - 5

Find the bisector of the angle between the lines 2x+y-6 = 0 and 2x-4y+7 = 0 which contains the point (1,2).

Solution :

Consider a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 (c1, c2 0),

If a1h+b1k+c1 and a2h+b2k+c2 have the same sign, point (h,k) will lie in the angle bisector

Solution Cont.

Given lines can be rewritten as -2x-y+6 = 0 and 2x-4y+7 = 0.

Now -2(1)-(2)+6 > 0 and 2(1)-4(2)+7 > 0

Angle bisector containing (1,2) is

Class Exercise - 6

Find the equation of the straight line drawn through the point of intersection of the lines x+y = 4 and 2x-3y = 1 and perpendicular to the line cutting off intercepts 5 and 6 on the positive axes.

Solution :

Family of lines through the intersection of the given lines is (x+y-4)+(2x-3y-1) = 0

Line cutting intercepts of 5 and 6 on the positive axes is

Solution Cont.

Slope of line perpendicular to this line will be -5/6.

Thus required equation is

3(x+y-4)+11(2x-3y-1) = 0

Or, 25x-30y-23 = 0

Class Exercise - 7

Find the separate equations of the pair of lines represented by 12x2+5xy-28y2+19x+61y-21 = 0.

Solution :

Rewrite the equation as a quadratic in x, we get 12x2+(5y+19)x-(28y2-61y+21) = 0

Given line can be rewritten as

Class Exercise - 8Show that the lines joining the origin to the points common to x2+hxy-y2+gx+fy = 0 and fx-gy = are at right angles whatever be the value of .

Solution :

Using this to make the equation of the curve homogeneous, we get

Solution Cont.

Sum of coefficients of x2 and y2

= (+fg)-(+fg)

= 0

Thus the required lines are perpendicular to each other, whatever be the value of .

Q.E.D.

Class Exercise - 9

Find the angle between the pair of lines : (x2+y2)sin2 = (xcos-ysin)2.

Solution :

Given equation can be rewritten as

(cos2-sin2)x2-2sincosxy+sin2-sin2)y2 = 0

Class Exercise - 10

Prove that the lines a2x2+2h(a+b)xy+b2y2 = 0 are equally inclined to the lines ax2+2hxy+by2 = 0.

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