Find the equations of the straight lines, bisecting the angles between the pair of straight lines 2 x + y = 4 and y + 3 x = 5 , placing first the bisector of the angle in which the origin lies.

The equations of the lines are given as, 2 x + y - 4 = 0 y + 3 x - 5 = 0 Now, making the constant term positive we get, - 2 x - y + 4 = 0 - 3 x - y + 5 = 0 The general equation of the angle bisectors to the lines a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 is given as, a 1 x + b 1 y + c 1 a 1 2 + b 1 2 = ± a 2 x + b 2 y + c 2 a 2 2 + b 2 2 Hence, the equation of the given angle bisector can be written as, - 2 x - y + 4 - 2 2 + - 1 2 = ± - 3 x - y + 5 - 3 2 + - 1 2 ⇒ - 2 x - y + 4 5 = ± - 3 x - y + 5 10 Now, the origin lies in the bisector, when we consider the positive sign in the above equation. This can be written as, ⇒ 2 - 2 x - y + 4 = - 3 x - y + 5 ⇒ 2 2 - 3 x + 2 - 1 y + 5 - 4 2 = 0 Considering the negative sign, we get, 2 - 2 x - y + 4 = - - 3 x - y + 5 ⇒ 2 2 + 3 x + 2 + 1 y - 5 + 4 2 = 0 Hence, the equations of the bisectors are, 2 2 - 3 x + 2 - 1 y + 5 - 4 2 = 0 2 2 + 3 x + 2 + 1 - 5 + 4 2 = 0

Find the bisectors of the angles between the straight lines y - b = 2 m 1 - m 2 x - a and y - b = 2 m ' 1 - m ' 2 x - a .

The equations of the lines are given as, y - b = 2 m 1 - m 2 x - a y - b = 2 m ' 1 - m ' 2 x - a Now, let m be tan α and m ' be tan β . Then, we get the equations of the lines as, y - b = 2 tan α 1 - tan 2 α x - a ⇒ y - b = tan 2 α x - a . . . i And, y - b = 2 tan β 1 - tan 2 β x - a ⇒ y - b = tan 2 β x - a . . . i i The lines are given by equations i and i i and they pass through the point a , b and have their angle of inclination as 2 α and 2 β , respectively. So, by geometry (as shown in the above diagram), the bisectors also pass through the point a , b and have inclination angles as α + β and π 2 + α + β . Hence, the equations of the bisectors are given as, B 1 : y - b = tan α + β x - a ⇒ y - b = tan α + tan β 1 - tan α . tan β x - a ⇒ y - b = m + m ' 1 - m m ' x - a ⇒ y - b 1 - m m ' - x - a m + m ' = 0 And, B 2 : y - b = tan π 2 + α + β x - a ⇒ y - b = - c o t α + β ( x - a ) ⇒ y - b tan α + β + ( x - a ) = 0 ⇒ ( y - b ) tan α + tan β 1 - tan α . tan β + x - a = 0 ⇒ ( y - b ) m + m ' 1 - m m ' + x - a = 0 ⇒ ( y - b ) ( m + m ' ) + ( x - a ) ( 1 - m m ' ) = 0 Hence, the equations of the bisectors are, y - b m + m ' + x - a 1 - m m ' = 0 y - b 1 - m m ' - x - a m + m ' = 0

S L Loney Solutions for Chapter: The Straight Line, Rectangular Coordinates, Exercise 4: EXAMPLES VIII

Author:S L Loney

S L Loney Mathematics Solutions for Exercise - S L Loney Solutions for Chapter: The Straight Line, Rectangular Coordinates, Exercise 4: EXAMPLES VIII

Attempt the practice questions on Chapter 3: The Straight Line, Rectangular Coordinates, Exercise 4: EXAMPLES VIII with hints and solutions to strengthen your understanding. The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates solutions are prepared by Experienced Embibe Experts.

Questions from S L Loney Solutions for Chapter: The Straight Line, Rectangular Coordinates, Exercise 4: EXAMPLES VIII with Hints & Solutions

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 38

Find the equations to the straight lines, bisecting the angles between the pairs of straight lines $12 x + 5 y - 4 = 0$ and $3 x + 4 y + 7 = 0$ , placing first the bisector of the angle in which the origin lies.

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 39

Find the equations of the straight lines, bisecting the angles between the pair of straight lines $4 x + 3 y - 7 = 0$ and $24 x + 7 y - 31 = 0$ , placing first the bisector of the angle in which the origin lies.

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 40

Find the equations of the straight lines, bisecting the angles between the pair of straight lines $2 x + y = 4$ and $5 (- 12 x - 5 y + 4) = 13 (3 x + 4 y + 7)$ , placing first the bisector of the angle in which the origin lies.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 41

Find the bisectors of the angles between the straight lines $y - b = \frac{2 m}{1 - m^{2}} (x - a)$ and $y - b = \frac{2 m^{'}}{1 - {m^{'}}^{2}} (x - a)$ .

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 42

Find the equations of the bisectors of the internal angles of the triangle the equations of whose sides are respectively $3 x + 4 y = 6, 12 x - 5 y = 3$ and $4 x - 3 y + 12 = 0$ .

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 43

Find the equations of the bisectors of the internal angles of the triangle the equations of it's sides are respectively $3 x + 5 y = 15, x + y = 4$ and $2 x + y = 6$ .

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 44

Find the equations to the straight lines passing through the foot of the perpendicular from the point $(h, k)$ upon the straight line $A x + B y + C = 0$ and bisecting the angles between the perpendicular and the given straight line.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 3 - The Straight Line, Rectangular Coordinates>EXAMPLES VIII>Q 45

Find the direction in which a straight line must be drawn through the point $(1, 2)$ , so that its point of intersection with the line $x + y = 4$ may be at a distance of $\frac{1}{3} \sqrt{6}$ units from this point.

S L Loney Solutions for Chapter: The Straight Line, Rectangular Coordinates, Exercise 4: EXAMPLES VIII

S L Loney Mathematics Solutions for Exercise - S L Loney Solutions for Chapter: The Straight Line, Rectangular Coordinates, Exercise 4: EXAMPLES VIII

Questions from S L Loney Solutions for Chapter: The Straight Line, Rectangular Coordinates, Exercise 4: EXAMPLES VIII with Hints & Solutions

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