Mathematics · Textbook solutions

Pair of Straight Lines

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 145 questions

4.1 Combined Equation of a Pair of Lines

4 q

4.1 Solved Ex.1 $x + y - 2 = 0$
4.1 Solved Ex.2 $x - 2 = 0$
4.1 Solved Ex.3 $x - 2y = 0$
4.1 Solved Ex.4 $x^2 - y^2 + x + y = 0$

4.2 Homogeneous Equation of Degree Two

37 q

Solved Examples

Worked · 12

4.1 Solved (4.2) Ex.1 $2x + 3y = 0$
4.1 Solved (4.2) Ex.1b $x^2 - 2xy - 3y^2 = 0$
4.1 Solved (4.2) Ex.2 $x^2 - 6xy + 9y^2 = 0$
4.1 Solved (4.2) Ex.3 $x^2 + 4xy - 7y^2 = 0$
Find the separate equations of lines represented by:
4.1 Solved (4.2) Ex.4 (i) $x^2 - 4y^2 = 0$
4.1 Solved (4.2) Ex.4 (ii) $3x^2 - 7xy + 4y^2 = 0$
4.1 Solved (4.2) Ex.4 (iii) $x^2 + 2xy - y^2 = 0$
4.1 Solved (4.2) Ex.4 (iv) $5x^2 - 3y^2 = 0$
4.1 Solved (4.2) Ex.5 $k$
4.1 Solved (4.2) Ex.6 $3x - 2y = 0$
4.1 Solved (4.2) Ex.7 $3x^2 + 2xy - y^2 = 0$
4.1 Solved (4.2) Ex.8 $k$

Exercise 4.1

Practice · 25

Find the combined equation of the following pairs of lines:
Q.1 (i) $2x + y = 0$
Q.1 (ii) $x + 2y - 1 = 0$
Q.1 (iii) $(2, 3)$
Q.1 (iv) $(2, 3)$
Q.1 (v) $(-1, 2)$
Find the separate equations of the lines represented by following equations:
Q.2 (i) $3y^2 + 7xy = 0$
Q.2 (ii) $5x^2 - 9y^2 = 0$
Q.2 (iii) $x^2 - 4xy = 0$
Q.2 (iv) $3x^2 - 10xy - 8y^2 = 0$
Q.2 (v) $3x^2 - 2\sqrt{3}\,xy - 3y^2 = 0$
Q.2 (vi) $x^2 + 2(\csc \alpha)xy + y^2 = 0$
Q.2 (vii) $x^2 + 2xy \tan \alpha - y^2 = 0$
Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by following equations:
Q.3 (i) $5x^2 - 8xy + 3y^2 = 0$
Q.3 (ii) $5x^2 + 2xy - 3y^2 = 0$
Q.3 (iii) $xy + y^2 = 0$
Q.3 (iv) $3x^2 - 4xy = 0$
Find $k$ if,
Q.4 (i) $x^2 + kxy - 3y^2 = 0$
Q.4 (ii) $3x^2 + kxy - y^2 = 0$
Q.4 (iii) $kx^2 + 4xy - y^2 = 0$
Find the condition that:
Q.5 (i) $4x + 5y = 0$
Q.5 (ii) $3x + y = 0$
Q.6 $ax^2 + 2hxy + by^2 = 0$
Q.7 $y = 3$
Q.8 $ax^2 + 2hxy + by^2 = 0$
Q.9 $ax^2 + 2hxy + by^2 = 0$

4.3 Angle between a Pair of Lines

18 q

Solved Examples

Worked · 9

4.2 Ex.1 $3x^2 - 4xy - 3y^2 = 0$
4.2 Ex.2 $x^2 + 4xy + 4y^2 = 0$
Find the acute angle between lines represented by:
i) $x^2 + xy = 0$
ii) $x^2 - 4xy + y^2 = 0$
iii) $3x^2 + 2xy - y^2 = 0$
iv) $2x^2 - 6xy + y^2 = 0$
v) $xy + y^2 = 0$
4.2 Ex.4 $\dfrac{\pi}{6}$
4.2 Ex.5 $60^\circ$

Exercise 4.2

Practice · 9

4.2 Q2 $x^2 + 6xy + 9y^2 = 0$
4.2 Q3 $k$
Find the measure of the acute angle between the lines represented by:
4.2 Q4 i) $3x^2 - 4\sqrt{3}\,xy + 3y^2 = 0$
4.2 Q4 ii) $4x^2 + 5xy + y^2 = 0$
4.2 Q4 iii) $2x^2 + 7xy + 3y^2 = 0$
4.2 Q4 iv) $(a^2 - 3b^2)x^2 + 8abxy + (b^2 - 3a^2)y^2 = 0$
4.2 Q5 $30^\circ$
4.2 Q6 $ax^2 + 2hxy + by^2 = 0$
4.2 Q7 $60^\circ$

4.4 General Second Degree Equation in x and y

19 q

Solved Examples

Worked · 4

4.3 Ex.1 $x^2 - 6xy + 5y^2 + 10x - 14y + 9 = 0$
4.3 Ex.2 $k$
4.3 Ex.3 $p$
4.3 Ex.4 $\triangle OAB$

Exercise 4.3

Practice · 15

Find the joint equation of the pair of lines:
4.3 Q1 i) $(2, -1)$
4.3 Q1 ii) $(2, -3)$
4.3 Q2 $x^2 + 2xy + 2y^2 + 2x + 2y + 1 = 0$
4.3 Q3 $2x^2 - xy - 3y^2 - 6x + 19y - 20 = 0$
4.3 Q4 $2x^2 + xy - y^2 + x + 4y - 3 = 0$
Find the separate equation of the lines represented by the following equations:
4.3 Q5 i) $(x - 2)^2 - 3(x - 2)(y + 1) + 2(y + 1)^2 = 0$
4.3 Q5 ii) $10(x + 1)^2 + (x + 1)(y - 2) - 3(y - 2)^2 = 0$
Find the value of $k$ if the following equations represent a pair of lines:
4.3 Q6 i) $3x^2 + 10xy + 3y^2 + 16y + k = 0$
4.3 Q6 ii) $kxy + 10x + 6y + 4 = 0$
4.3 Q6 iii) $x^2 + 3xy + 2y^2 + x - y + k = 0$
4.3 Q7 $p$
4.3 Q8 $p$
4.3 Q9 $x^2 - 7x + 6 = 0$
4.3 Q10 $\triangle OAB$
4.3 Q11 $x^2 - y^2 - 2x + 1 = 0$

Miscellaneous Exercise 4

67 q

Choose the correct alternative

Practice · 14

Misc I (1) $4x^2 + hxy + y^2 = 0$
Tap an option to check your answer.
Misc I (2) $kx^2 - 3xy + 6y^2 = 0$
Tap an option to check your answer.
Misc I (3) $2x^2 + 3xy - 9y^2 = 0$
Tap an option to check your answer.
Misc I (4) $3x^2 - 4xy + y^2 = 0$
Tap an option to check your answer.
Misc I (5) $ax^2 + 2hxy + by^2 = 0$
Tap an option to check your answer.
Misc I (6) $\dfrac{x^2}{a} + \dfrac{2xy}{h} + \dfrac{y^2}{b} = 0$
Tap an option to check your answer.
Misc I (7) $3x^2 + 4xy - 5y^2 = 0$
Tap an option to check your answer.
Misc I (8) $ax^2 + 2hxy + by^2 = 0$
Tap an option to check your answer.
Misc I (9) $3x^2 - 8xy + qy^2 + 2x + 14y + p = 1$
Tap an option to check your answer.
Misc I (10) $x^2 + 4xy + y^2 = 0$
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Misc I (11) $The combined equation of the co-ordinate axes is __________.$
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Misc I (12) $h^2 = ab$
Tap an option to check your answer.
Misc I (13) $ax^2 + 2hxy + by^2 = 0$
Tap an option to check your answer.
Misc I (14) $(x - 2y)^2 + k(x - 2y) = 0$
Tap an option to check your answer.

Solve the following

Practice · 53

Find the joint equation of lines:
Misc II Q.1 (i) $x - y = 0$
Misc II Q.1 (ii) $x + y - 3 = 0$
Misc II Q.1 (iii) $Passing through the origin and having slopes 2 and 3.$
Misc II Q.1 (iv) $60^\circ$
Misc II Q.1 (v) $(1, 2)$
Misc II Q.1 (vi) $(3, 2)$
Misc II Q.1 (vii) $(-1, 2)$
Misc II Q.1 (viii) $1 + \sqrt{3}$
Misc II Q.1 (ix) $Which are at a distance of 9 units from the Y-axis.$
Misc II Q.1 (x) $(3, 2)$
Misc II Q.1 (xi) $x + 2y = 19$
Show that each of the following equation represents a pair of lines.
Misc II Q.2 (i) $x^2 + 2xy - y^2 = 0$
Misc II Q.2 (ii) $4x^2 + 4xy + y^2 = 0$
Misc II Q.2 (iii) $x^2 - y^2 = 0$
Misc II Q.2 (iv) $x^2 + 7xy - 2y^2 = 0$
Misc II Q.2 (v) $x^2 - 2\sqrt{3}\,xy - y^2 = 0$
Find the separate equations of lines represented by the following equations:
Misc II Q.3 (i) $6x^2 - 5xy - 6y^2 = 0$
Misc II Q.3 (ii) $x^2 - 4y^2 = 0$
Misc II Q.3 (iii) $3x^2 - y^2 = 0$
Misc II Q.3 (iv) $2x^2 + 2xy - y^2 = 0$
Find the joint equation of the pair of lines through the origin and perpendicular to the lines given by:
Misc II Q.4 (i) $x^2 + 4xy - 5y^2 = 0$
Misc II Q.4 (ii) $2x^2 - 3xy - 9y^2 = 0$
Misc II Q.4 (iii) $x^2 + xy - y^2 = 0$
Find $k$ if:
Misc II Q.5 (i) $3x^2 + kxy - y^2 = 0$
Misc II Q.5 (ii) $2x^2 + kxy - 3y^2 = 0$
Misc II Q.5 (iii) $3x^2 - 4xy + ky^2 = 0$
Misc II Q.5 (iv) $3x^2 - kxy + 5y^2 = 0$
Misc II Q.5 (v) $3x^2 + 4xy + ky^2 = 0$
Misc II Q.5 (vi) $kx^2 + 5xy + y^2 = 0$
Misc II Q.5 (vii) $6x^2 + kxy + y^2 = 0$
Misc II Q.6 $x^2 + 3xy + 2y^2 = 0$
Misc II Q.7 $x = 3$
Misc II Q.8 $x^2 - 4xy + y^2 = 0$
Misc II Q.9 $ax^2 + 2hxy + by^2 = 0$
Misc II Q.10 $5x^2 + 6xy - y^2 = 0$
Misc II Q.11 $a$
Misc II Q.12 $4x - 5y = 0$
Show that the following equations represent a pair of lines, find the acute angle between them.
Misc II Q.13 (i) $9x^2 - 6xy + y^2 + 18x - 6y + 8 = 0$
Misc II Q.13 (ii) $2x^2 + xy - y^2 + x + 4y - 3 = 0$
Misc II Q.13 (iii) $(x - 3)^2 + (x - 3)(y - 4) - 2(y - 4)^2 = 0$
Misc II Q.14 $60^\circ$
Misc II Q.15 $ax^2 + 2hxy + by^2 = 0$
Misc II Q.16 $\alpha$
Misc II Q.17 $3x + 4y + 5 = 0$
Misc II Q.18 $x^2 - 4xy + y^2 = 0$
Misc II Q.19 $ax^2 + 2hxy + by^2 = 0$
Misc II Q.20 $P(x_1, y_1)$
Misc II Q.21 $(\tan^2\theta + \cos^2\theta)x^2 - 2xy\tan\theta + (\sin^2\theta)y^2 = 0$
Misc II Q.22 $ay^2 + bxy + ex + dy = 0$
Misc II Q.23 $ax^2 + 2hxy + by^2 = 0$
Misc II Q.24 $x + 2 = 0$
Misc II Q.25 $ax^2 + 2hxy + by^2 = 0$
Misc II Q.26 $ax^2 - y^2 + 2y + c = 1$

What the board has asked from this chapter

Questions per paper · 14 papers · 2015 to 2026

June 2026

2 q

Q. 6 $k$
Combined Equation of a Pair of Lines
Q. 28 $x$
Combined Equation of a Pair of Lines

February 2026

2 q

Q. 6 $k$
Combined Equation of a Pair of Lines
Q. 28 $x$
Combined Equation of a Pair of Lines

July 2025

2 q

Q. 16 $3x + 2y - 1 = 0$
Combined Equation of a Pair of Lines
Q. 17 $\theta$
Angle between a Pair of Lines

2025

2 q

Q. 28 $x$
Combined Equation of a Pair of Lines
Q. 6 $x^{2} - y^{2} - 2x + 1 = 0$
General Second Degree Equation of Two Lines

July 2024

3 q

Q. 2. (i) $2x + y = 0$
Combined Equation of a Pair of Lines
Q. 6 $xy + y^{2} = 0$
Angle between a Pair of Lines
Q. 17 $(\tan^{2}\theta + \cos^{2}\theta)x^{2} - 2xy\tan\theta + (\sin^{2}\theta)y^{2} = 0$
Combined Equation of a Pair of Lines

2024

2 q

Q. 6 $k,$
Combined Equation of a Pair of Lines
Q. 28 $\theta$
Angle between a Pair of Lines

March 2023

3 q

Q. 6 $ax^{2} + 2hxy + by^{2} = 0$
Combined Equation of a Pair of Lines
Q. 17 $\theta$
Angle between a Pair of Lines
Q. 2. i. $Write the joint equation of co-ordinate axes.$
Combined Equation of a Pair of Lines

March 2022

3 q

Q. 16 $ax^{2} + 2hxy + by^{2} = 0$
Angle between a Pair of Lines
Q. 2. ii. $x^{2} - 9y^{2} = 0$
Combined Equation of a Pair of Lines
Q. 6 $k,$
Combined Equation of a Pair of Lines

February 2020

2 q

Q. 27 $x$
Combined Equation of a Pair of Lines
Q. 5 $3x^{2} + 4xy - 3y^{2} = 0$
Angle between a Pair of Lines

March 2019

1 q

Q. 24 $px^{2} - 8xy + 3y^{2} + 14x + 2y + q = 0$
General Second Degree Equation of Two Lines

March 2018

2 q

Q. 2. (B) i. $x$
Combined Equation of a Pair of Lines
Q. 3. (A) i. $5x^{2} + 2xy - 3y^{2} = 0$
Combined Equation of a Pair of Lines

March 2017

2 q

Q. 2. (B) ii. $\theta$
Angle between a Pair of Lines
Q. 3. (A) ii. $5x^{2} + 2xy - 3y^{2} = 0$
Combined Equation of a Pair of Lines

March 2016

1 q

Q. 3. (B) iii. $x$
Combined Equation of a Pair of Lines

March 2015

3 q

Q. 1. (A) iii. $hxy + gx + fy + c = 0$
Tap an option to check your answer.
General Second Degree Equation of Two Lines
Q. 1. (B) ii. $x^{2} + kxy - 3y^{2} = 0$
Combined Equation of a Pair of Lines
Q. 3. (A) i. $\theta$
Angle between a Pair of Lines

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 6 times: Mar 2016, Mar 2018, Feb 2020, Feb 2025, Feb 2026, Jun 2026
Q. 28 $x$
Question 2
Asked 5 times: Mar 2015, Mar 2017, Mar 2023, Mar 2024, Jul 2025
Q. 17 $\theta$
Question 3
Asked 3 times: Mar 2015, Mar 2024, Feb 2026
Q. 6 $k$
Question 4
Asked 2 times: Mar 2022, Jun 2026
Q. 6 $k$
Question 5
Asked 2 times: Mar 2017, Mar 2018
Q. 3. (A) i. $5x^{2} + 2xy - 3y^{2} = 0$
Question 6
This type asked 3 times: Mar 2023, Jul 2024, Jul 2025. This one: Jul 2025
Q. 16 $3x + 2y - 1 = 0$
Question 7
This type asked 2 times: Feb 2020, Jul 2024. This one: Jul 2024
Q. 6 $xy + y^{2} = 0$
Question 8
Asked once: Jul 2024 (same type asked 3 times)
Q. 2. (i) $2x + y = 0$
Question 9
Asked once: Feb 2020 (same type asked 2 times)
Q. 5 $3x^{2} + 4xy - 3y^{2} = 0$
Question 10
Asked once: Mar 2023 (same type asked 3 times)
Q. 2. i. $Write the joint equation of co-ordinate axes.$

4.1 Combined Equation of a Pair of Lines

4.2 Homogeneous Equation of Degree Two

Solved Examples

Exercise 4.1

4.3 Angle between a Pair of Lines

Solved Examples

Exercise 4.2

4.4 General Second Degree Equation in x and y

Solved Examples

Exercise 4.3

Miscellaneous Exercise 4

Choose the correct alternative

Solve the following

What the board has asked from this chapter

June 2026

February 2026

July 2025

2025

July 2024

2024

March 2023

March 2022

February 2020

March 2019

March 2018

March 2017

March 2016

March 2015

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10