#### Chapters

## Chapter 4: Pair of Straight Lines

#### Exercise 4.1 [Pages 119 - 120]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight Lines Exercise 4.1 [Pages 119 - 120]

**Find the combined equation of the following pair of line:**

2x + y = 0 and 3x - y = 0

**Find the combined equation of the following pair of line:**

x + 2y - 1 = 0 and x - 3y + 2 = 0

**Find the combined equation of the following pair of line:**

passing through (2, 3) and parallel to the coordinate axes.

**Find the combined equation of the following pair of line:**

passing through (2, 3) and perpendicular to the lines 3x + 2y - 1 = 0 and x - 3y + 2 = 0

**Find the combined equation of the following pair of line:**

passing through (-1, 2), one is parallel to x + 3y - 1 = 0 and other is perpendicular to 2x - 3y - 1 = 0

**Find the separate equation of the line represented by the following equation:**

3y^{2} + 7xy = 0

**Find the separate equation of the line represented by the following equation:**

5y^{2} + 9y^{2} = 0

**Find the separate equation of the line represented by the following equation:**

x^{2} - 4xy = 0

Find the separate equations of the lines represented by the equation ` 3"x"^2-10"xy"-8"y"^2=0`

**Find the separate equation of the line represented by the following equation:**

`3"x"^2 - 2sqrt3"xy" - 3"y"^2 = 0`

**Find the separate equation of the line represented by the following equation:**

x^{2} + 2(cosec α)xy + y^{2} = 0

**Find the separate equation of the line represented by the following equation:**

x^{2} + 2xy tan α - y^{2} = 0

**Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by following equation:**

5x^{2} - 8xy + 3y^{2} = 0

**Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:**

5x^{2} + 2xy - 3y^{2} = 0

**Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:**

xy + y^{2} = 0

**Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:**

3x^{2} - 4xy = 0

Find k, if the sum of the slopes of the lines represented by x^{2} + kxy - 3y^{2} = 0 is twice their product.

Find k, the slopes of the lines represented by 3x^{2} + kxy - y^{2} = 0 differ by 4.

Find k, the slope of one of the lines given by kx^{2} + 4xy - y^{2} = 0 exceeds the slope of the other by 8.

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax^{2} + 2hxy + by^{2} = 0

Find the condition that the line 3x + y = 0 may be perpendicular to one of the lines given by ax^{2} + 2hxy + by^{2} = 0

If one of the lines given by ax^{2} + 2hxy + by^{2} = 0 is perpendicular to px + qy = 0, show that ap^{2} + 2hpq + bq^{2} = 0.

Find the combined equation of the pair of lines through the origin and making an equilateral triangle with the line y = 3.

If the slope of one of the lines given by ax^{2} + 2hxy + by^{2} = 0 is four times the other, show that 16h^{2} = 25ab.

If one of the lines given by ax^{2} + 2hxy + by^{2} = 0 bisect an angle between the coordinate axes, then show that (a + b)^{2} = 4h^{2} .

#### Exercise 4.2 [Page 124]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight Lines Exercise 4.2 [Page 124]

. Show that the lines represented by 3x^{2} - 4xy - 3y^{2} = 0 are perpendicular to each other.

Show that the lines represented by x^{2} + 6xy + 9y^{2} = 0 are coincident.

Find the value of k if lines represented by kx^{2} + 4xy - 4y^{2} = 0 are perpendicular to each other.

**Find the measure of the acute angle between the line represented by:**

`3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0`

**Find the measure of the acute angle between the line represented by:**

4x^{2 }+ 5xy + y^{2} = 0

**Find the measure of the acute angle between the line represented by:**

2x^{2 }+ 7xy + 3y^{2} = 0

**Find the measure of the acute angle between the line represented by:**

(a^{2} + 3b^{2})x^{2} + 8abxy + (b^{2} - 3a^{2})y^{2} = 0

Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x + 2y - 11 = 0

If the angle between the lines represented by ax^{2} + 2hxy + by^{2} = 0 is equal to the angle between the lines 2x^{2} - 5xy + 3y^{2} = 0, then show that 100 (h^{2} - ab) = (a + b)^{2}.

Find the combined equation of lines passing through the origin and each of which making an angle of 60° with the Y-axis.

#### Exercise 4.3 [Pages 127 - 128]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight Lines Exercise 4.3 [Pages 127 - 128]

Find the joint equation of the pair of the line through the point (2, -1) and parallel to the lines represented by 2x^{2} + 3xy - 9y^{2} = 0.

Find the joint equation of the pair of the line through the point (2, -3) and parallel to the lines represented by x^{2} + xy - y^{2} = 0.

Show that the equation x^{2} + 2xy + 2y^{2} + 2x + 2y + 1 = 0 does not represent a pair of lines.

Show that the equation 2x^{2} - xy - 3y^{2} - 6x + 19y - 20 = 0 represents a pair of lines.

Show that the equation 2x^{2} + xy - y^{2} + x + 4y - 3 = 0 represents a pair of lines. Also, find the acute angle between them.

**Find the separate equation of the line represented by the following equation:**

(x - 2)^{2} - 3(x - 2)(y + 1) + 2(y + 1)^{2} = 0

**Find the separate equation of the line represented by the following equation:**

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 line:**

3x^{2} + 10xy + 3y^{2} + 16y + k = 0

**Find the value of k, if the following equations represent a pair of line:**

kxy + 10x + 6y + 4 = 0

**Find the value of k, if the following equations represent a pair of line:**

x^{2} + 3xy + 2y^{2} + x - y + k = 0

Find p and q, if the equation px^{2} - 8xy + 3y^{2} + 14x + 2y + q = 0 represents a pair of perpendicular lines.

Find p and q, if the equation 2x^{2} + 8xy + py^{2} + qx + 2y - 15 = 0 represents a pair of parallel lines.

Equations of pairs of opposite sides of a parallelogram are x^{2} - 7x + 6 = 0 and y^{2} - 14y + 40 = 0. Find the joint equation of its diagonals.

ΔOAB is formed by the lines x^{2} - 4xy + y^{2} = 0 and the line AB. The equation of line AB is 2x + 3y - 1 = 0. Find the equation of the median of the triangle drawn from O.

Find the coordinates of the points of intersection of the lines represented by x^{2} - y^{2} - 2x + 1 = 0

#### Miscellaneous Exercise 4 [Pages 129 - 130]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight Lines Miscellaneous Exercise 4 [Pages 129 - 130]

**Choose correct alternatives:**

If the equation 4x^{2} + hxy + y^{2} = 0 represents two coincident lines, then h = _______

± 2

± 3

± 4

± 5

**Choose correct alternatives:**

If the lines represented by kx^{2} - 3xy + 6y^{2} = 0 are perpendicular to each other, then

k = 6

k = - 6

k = 3

k = - 3

**Choose correct alternatives:**

Auxiliary equation of 2x^{2} + 3xy - 9y^{2} = 0 is

2m

^{2}+ 3m - 9 = 09m

^{2}- 3m - 2 = 02m

^{2}- 3m + 9 = 0- 9m

^{2}- 3m + 2 = 0

**Choose correct alternatives:**

The difference between the slopes of the lines represented by 3x^{2} - 4xy + y^{2} = 0 is 2

2

1

3

4

**Choose correct alternatives:**

If two lines ax^{2} + 2hxy + by^{2} = 0 make angles α and β with X-axis, then tan (α + β) = _____.

`"h"/("a + b")`

`"h"/("a - b")`

`"2h"/("a + b")`

`"2h"/("a - b")`

**Choose correct alternatives:**

If the slope of one of the two lines given by `"x"^2/"a" + "2xy"/"h" + "y"^2/"b" = 0` is twice that of the other, then ab : h^{2} = ______.

1 : 2

2 : 1

8 : 9

9 : 8

**Choose correct alternatives:**

The joint equation of the lines through the origin and perpendicular to the pair of lines 3x^{2} + 4xy - 5y^{2} = 0 is _______.

5x

^{2}+ 4xy - 3y^{2}= 03x

^{2}+ 4xy - 5y^{2}= 03x

^{2}- 4xy + 5y^{2}= 05x

^{2}+ 4xy + 3y^{2}= 0

**Choose correct alternatives:**

If acute angle between lines ax^{2} + 2hxy + by^{2} = 0 is, `pi/4`, then 4h^{2} = ______.

a

^{2}+ 4ab + b^{2}a

^{2}+ 6ab + b^{2}(a + 2b)(a + 3b)

(a - 2b)(2a + b)

**Choose correct alternatives:**

If the equation 3x^{2} - 8xy + qy^{2} + 2x + 14y + p = 1 represents a pair of perpendicular lines, then the values of p and q are respectively.

- 3 and - 7

- 7 and - 3

3 and 7

- 7 and 3

**Choose correct alternatives:**

The area of triangle formed by the lines x^{2} + 4xy + y^{2} = 0 and x - y - 4 = 0 is

`4/sqrt3` sq units

`8/sqrt3` sq units

`16/sqrt3` sq units

`15/sqrt3` sq units

**Choose correct alternatives:**

The combined equation of the coordinate axes is

x + y = 0

xy = k

xy = 0

x - y = k

**Choose correct alternatives:**

If h^{2} = ab, then slopes of lines ax^{2} + 2hxy + by^{2} = 0 are in the ratio

1:2

2:1

2:3

1:1

**Choose correct alternatives:**

If slope of one of the lines ax^{2} + 2hxy + by^{2} = 0 is 5 times the slope of the other, then 5h^{2} = ______

ab

2ab

7ab

9ab

**Choose correct alternatives:**

If distance between lines (x - 2y)^{2} + k(x - 2y) = 0 is 3 units, then k = ______.

± 3

± 5`sqrt5`

0

`±3sqrt5`

#### Miscellaneous Exercise 4 [Pages 130 - 132]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Pair of Straight Lines Miscellaneous Exercise 4 [Pages 130 - 132]

**Find the joint equation of the line:**

x - y = 0 and x + y = 0

**Find the joint equation of the line:**

x + y - 3 = 0 and 2x + y - 1 = 0

Find the joint equation of the line passing through the origin having slopes 2 and 3.

Find the joint equation of the line passing through the origin and having inclinations 60° and 120°.

Find the joint equation of the line passing through (1, 2) and parallel to the coordinate axes

Find the joint equation of the line passing through (3, 2) and parallel to the lines x = 2 and y = 3.

Find the joint equation of the line passing through (-1, 2) and perpendicular to x + 2y + 3 = 0 and 3x - 4y - 5 = 0

Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`

Find the joint equation of the line which are at a distance of 9 units from the Y-axis.

Find the joint equation of the line passing through the point (3, 2), one of which is parallel to the line x - 2y = 2, and other is perpendicular to the line y = 3.

Find the joint equation of the line passing through the origin and perpendicular to the lines x + 2y = 19 and 3x + y = 18

**Show that the following equations represents a pair of line:**

x^{2} + 2xy - y^{2} = 0

**Show that the following equations represents a pair of line:**

4x^{2} + 4xy + y^{2} = 0

**Show that the following equations represent a pair of line:**

x^{2} - y^{2} = 0

**Show that the following equations represent a pair of line:**

x^{2} + 7xy - 2y^{2} = 0

**Show that the following equations represent a pair of line:**

`"x"^2 - 2sqrt3"xy" - "y"^2 = 0`

**Find the separate equation of the line represented by the following equation:**

6x^{2} - 5xy - 6y^{2} = 0

**Find the separate equation of the line represented by the following equation:**

x^{2} - 4y^{2} = 0

**Find the separate equation of the line represented by the following equation:**

3x^{2} - y^{2} = 0

**Find the separate equation of the line represented by the following equation:**

2x^{2} + 2xy - y^{2} = 0

**Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by**

x^{2} + 4xy - 5y^{2} = 0

**Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by**

2x^{2} - 3xy - 9y^{2} = 0

**Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by**

x^{2} + xy - y^{2} = 0

Find k, if the sum of the slopes of the lines given by 3x^{2} + kxy - y^{2} = 0 is zero.

Find k, if the sum of the slopes of the lines given by x^{2} + kxy - 3y^{2} = 0 is equal to their product.

Find k, if the slope of one of the lines given by 3x^{2} - 4xy + ky^{2} = 0 is 1.

Find k, if one of the lines given by 3x^{2} - kxy + 5y^{2} = 0 is perpendicular to the line 5x + 3y = 0.

Find k if the slope of one of the lines given by 3x^{2 }+ 4xy + ky^{2} = 0 is three times the other.

Find k, if the slopes of lines given by kx^{2} + 5xy + y^{2} = 0 differ by 1.

Find k, if one of the lines given by 6x^{2} + kxy + y^{2} = 0 is 2x + y = 0.

Find the joint equation of the pair of lines which bisect angles between the lines given by x^{2} + 3xy + 2y^{2} = 0

Find the joint equation of the pair of lines through the origin and making an equilateral triangle with the line x = 3.

Show that the lines x^{2} - 4xy + y^{2} = 0 and x + y = 10 contain the sides of an equilateral triangle. Find the area of the triangle.

If the slope of one of the lines given by ax^{2} + 2hxy + by^{2} = 0 is three times the other, prove that 3h^{2} = 4ab.

Find the combined equation of bisectors of angles between the lines represented by 5x^{2} + 6xy - y^{2} = 0.

Find an if the sum of the slope of lines represented by ax^{2} + 8xy + 5y^{2} = 0 is twice their product.

If the line 4x - 5y = 0 coincides with one of the lines given by ax^{2} + 2hxy + by^{2} = 0, then show that 25a + 40h + 16b = 0

**Show that the following equation represents a pair of line. Find the acute angle between them:**

9x^{2} - 6xy + y^{2} + 18x - 6y + 8 = 0

**Show that the following equation represents a pair of line. Find the acute angle between them:**

2x^{2} + xy - y^{2} + x + 4y - 3 = 0

**Show that the following equation represents a pair of line. Find the acute angle between them:**

(x - 3)^{2} + (x - 3)(y - 4) - 2(y - 4)^{2} = 0

Find the combined equation of lines passing through the origin and each of which making an angle of 60° with the Y-axis.

If the lines represented by ax^{2} + 2hxy + by^{2} = 0 make angles of equal measure with the coordinate axes, then show that a ± b.

OR

Show that, one of the lines represented by ax^{2} + 2hxy + by^{2} = 0 will make an angle of the same measure with the X-axis as the other makes with the Y-axis, if a = ± b.

Show that the combined equation of the pair of lines passing through the origin and each making an angle α with the line x + y = 0 is x^{2} + 2(sec 2α)xy + y^{2} = 0

Show that the line 3x + 4y + 5 = 0 and the lines (3x + 4y)^{2} - 3(4x - 3y)^{2} = 0 form the sides of an equilateral triangle.

Show that the lines x^{2} - 4xy + y^{2} = 0 and the line x + y = `sqrt6` form an equilateral triangle. Find its area and perimeter.

If the slope of one of the lines given by ax^{2} + 2hxy + by^{2} = 0 is square of the slope of the other line, show that a^{2}b + ab^{2} + 8h^{3} = 6abh.

Prove that the product of length of perpendiculars drawn from P(x_{1}, y_{1}) to the lines represented by ax^{2} + 2hxy + by^{2} = 0 is `|("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a - b")^2 + "4h"^2)|`

Show that the difference between the slopes of the lines given by (tan^{2}θ + cos^{2}θ)x^{2} - 2xy tan θ + (sin^{2}θ)y^{2} = 0 is two.

Find the condition that the equation ay^{2} + bxy + ex + dy = 0 may represent a pair of lines.

If the lines given by ax^{2} + 2hxy + by^{2} = 0 form an equilateral triangle with the line lx + my = 1, show that (3a + b)(a + 3b) = 4h^{2}.

If the line x + 2 = 0 coincides with one of the lines represented by the equation x^{2} + 2xy + 4y + k = 0, then prove that k = - 4.

Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax^{2} + 2hxy + by^{2} = 0 is bx^{2} - 2hxy + ay^{2} = 0.

If equation ax^{2} - y^{2} + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.

## Chapter 4: Pair of Straight Lines

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 - Pair of Straight Lines

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 (Pair of Straight Lines) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 Pair of Straight Lines are Combined Equation of a Pair Lines, Homogeneous Equation of Degree Two, Angle Between Lines, General Second Degree Equation, Equation of a Line in Space.

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