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

## Chapter 4: Pair of Straight Lines

Exercise 4.1Exercise 4.2Exercise 4.3Miscellaneous Exercise 4

### 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]

Exercise 4.1 | Q 1.1 | Page 119

Find the combined equation of the following pair of line:

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

Exercise 4.1 | Q 1.2 | Page 119

Find the combined equation of the following pair of line:

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

Exercise 4.1 | Q 1.3 | Page 119

Find the combined equation of the following pair of line:

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

Exercise 4.1 | Q 1.4 | Page 119

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

Exercise 4.1 | Q 1.5 | Page 119

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

Exercise 4.1 | Q 2.1 | Page 119

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

3y2 + 7xy = 0

Exercise 4.1 | Q 2.2 | Page 119

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

5y2 + 9y2 = 0

Exercise 4.1 | Q 2.3 | Page 119

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

x2 - 4xy = 0

Exercise 4.1 | Q 2.4 | Page 119

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

Exercise 4.1 | Q 2.5 | Page 119

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

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

Exercise 4.1 | Q 2.6 | Page 119

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

x2 + 2(cosec α)xy + y2 = 0

Exercise 4.1 | Q 2.7 | Page 119

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

x2 + 2xy tan α - y2 = 0

Exercise 4.1 | Q 3.1 | Page 119

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

5x2 - 8xy + 3y2 = 0

Exercise 4.1 | Q 3.2 | Page 119

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

5x2 + 2xy - 3y2 = 0

Exercise 4.1 | Q 3.3 | Page 119

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 + y2 = 0

Exercise 4.1 | Q 3.4 | Page 119

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

3x2 -  4xy = 0

Exercise 4.1 | Q 4.1 | Page 119

Find k, if the sum of the slopes of the lines represented by x2 + kxy - 3y2 = 0 is twice their product.

Exercise 4.1 | Q 4.2 | Page 119

Find k, the slopes of the lines represented by 3x2 + kxy - y2 = 0 differ by 4.

Exercise 4.1 | Q 4.3 | Page 119

Find k, the slope of one of the lines given by kx2 + 4xy - y2 = 0 exceeds the slope of the other by 8.

Exercise 4.1 | Q 5.1 | Page 120

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0

Exercise 4.1 | Q 5.2 | Page 120

Find the condition that the line 3x + y = 0 may be perpendicular to one of the lines given by ax2 + 2hxy + by2 = 0

Exercise 4.1 | Q 6 | Page 120

If one of the lines given by ax2 + 2hxy + by2 = 0 is perpendicular to px + qy = 0, show that ap2 + 2hpq + bq2 = 0.

Exercise 4.1 | Q 7 | Page 120

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

Exercise 4.1 | Q 8 | Page 120

If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is four times the other, show that 16h2 = 25ab.

Exercise 4.1 | Q 9 | Page 120

If one of the lines given by ax2 + 2hxy + by2 = 0 bisect an angle between the coordinate axes, then show that (a + b)2 = 4h2 .

### 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]

Exercise 4.2 | Q 1 | Page 124

. Show that the lines represented by 3x2 - 4xy - 3y2 = 0 are perpendicular to each other.

Exercise 4.2 | Q 2 | Page 124

Show that the lines represented by x2 + 6xy + 9y2 = 0 are coincident.

Exercise 4.2 | Q 3 | Page 124

Find the value of k if lines represented by kx2 + 4xy - 4y2 = 0 are perpendicular to each other.

Exercise 4.2 | Q 4.1 | Page 124

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

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

Exercise 4.2 | Q 4.2 | Page 124

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

4x2 + 5xy + y2 = 0

Exercise 4.2 | Q 4.3 | Page 124

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

2x2 + 7xy + 3y2 = 0

Exercise 4.2 | Q 4.4 | Page 124

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

(a2 + 3b2)x2 + 8abxy + (b2 - 3a2)y2 = 0

Exercise 4.2 | Q 5 | Page 124

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

Exercise 4.2 | Q 6 | Page 124

If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 = 0, then show that 100 (h2 - ab) = (a + b)2

Exercise 4.2 | Q 7 | Page 124

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

### 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]

Exercise 4.3 | Q 1.1 | Page 127

Find the joint equation of the pair of the line through the point (2, -1) and parallel to the lines represented by 2x2 + 3xy - 9y2 = 0.

Exercise 4.3 | Q 1.2 | Page 127

Find the joint equation of the pair of the line through the point (2, -3) and parallel to the lines represented by x2 + xy - y2 = 0.

Exercise 4.3 | Q 2 | Page 127

Show that the equation x2 + 2xy + 2y2 + 2x + 2y + 1 = 0 does not represent a pair of lines.

Exercise 4.3 | Q 3 | Page 127

Show that the equation 2x2 - xy - 3y2 - 6x + 19y - 20 = 0 represents a pair of lines.

Exercise 4.3 | Q 4 | Page 127

Show that the equation 2x2 + xy - y2 + x + 4y - 3 = 0 represents a pair of lines. Also, find the acute angle between them.

Exercise 4.3 | Q 5.1 | Page 127

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

Exercise 4.3 | Q 5.2 | Page 127

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

Exercise 4.3 | Q 6.1 | Page 127

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

3x2 + 10xy + 3y2 + 16y + k = 0

Exercise 4.3 | Q 6.2 | Page 127

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

kxy + 10x + 6y + 4 = 0

Exercise 4.3 | Q 6.3 | Page 127

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

x2 + 3xy + 2y2 + x - y + k = 0

Exercise 4.3 | Q 7 | Page 128

Find p and q, if the equation px2 - 8xy + 3y2 + 14x + 2y + q = 0 represents a pair of perpendicular lines.

Exercise 4.3 | Q 8 | Page 128

Find p and q, if the equation 2x2 + 8xy + py2 + qx + 2y - 15 = 0 represents a pair of parallel lines.

Exercise 4.3 | Q 9 | Page 128

Equations of pairs of opposite sides of a parallelogram are x2 - 7x + 6 = 0 and y2 - 14y + 40 = 0. Find the joint equation of its diagonals.

Exercise 4.3 | Q 10 | Page 128

ΔOAB is formed by the lines x2 - 4xy + y2 = 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.

Exercise 4.3 | Q 11 | Page 128

Find the coordinates of the points of intersection of the lines represented by x2 - y2 - 2x + 1 = 0

### 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]

Miscellaneous Exercise 4 | Q 1.01 | Page 129

Choose correct alternatives:

If the equation 4x2 + hxy + y2 = 0 represents two coincident lines, then h = _______

• ± 2

• ± 3

• ± 4

• ± 5

Miscellaneous Exercise 4 | Q 1.02 | Page 129

Choose correct alternatives:

If the lines represented by kx2 - 3xy + 6y2 = 0 are perpendicular to each other, then

• k = 6

• k = - 6

• k = 3

• k = - 3

Miscellaneous Exercise 4 | Q 1.03 | Page 129

Choose correct alternatives:

Auxiliary equation of 2x2 + 3xy - 9y2 = 0 is

• 2m2 + 3m - 9 = 0

• 9m2 - 3m - 2 = 0

• 2m2 - 3m + 9 = 0

• - 9m2 - 3m + 2 = 0

Miscellaneous Exercise 4 | Q 1.04 | Page 129

Choose correct alternatives:

The difference between the slopes of the lines represented by 3x2 - 4xy + y2 = 0 is 2

• 2

• 1

• 3

• 4

Miscellaneous Exercise 4 | Q 1.05 | Page 129

Choose correct alternatives:

If two lines ax2 + 2hxy + by2 = 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")

Miscellaneous Exercise 4 | Q 1.06 | Page 129

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 : h2 = ______.

• 1 : 2

• 2 : 1

• 8 : 9

• 9 : 8

Miscellaneous Exercise 4 | Q 1.07 | Page 130

Choose correct alternatives:

The joint equation of the lines through the origin and perpendicular to the pair of lines 3x2 + 4xy - 5y2 = 0 is _______.

• 5x2 + 4xy - 3y2 = 0

• 3x2 + 4xy - 5y2 = 0

• 3x2 - 4xy + 5y2 = 0

• 5x2 + 4xy + 3y2 = 0

Miscellaneous Exercise 4 | Q 1.08 | Page 130

Choose correct alternatives:

If acute angle between lines ax2 + 2hxy + by2 = 0 is, pi/4, then 4h2 = ______.

• a2 + 4ab + b2

• a2 + 6ab + b2

• (a + 2b)(a + 3b)

• (a - 2b)(2a + b)

Miscellaneous Exercise 4 | Q 1.09 | Page 130

Choose correct alternatives:

If the equation 3x2 - 8xy + qy2 + 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

Miscellaneous Exercise 4 | Q 1.1 | Page 130

Choose correct alternatives:

The area of triangle formed by the lines x2 + 4xy + y2 = 0 and x - y - 4 = 0 is

• 4/sqrt3 sq units

• 8/sqrt3 sq units

• 16/sqrt3 sq units

• 15/sqrt3 sq units

Miscellaneous Exercise 4 | Q 1.11 | Page 130

Choose correct alternatives:

The combined equation of the coordinate axes is

• x + y = 0

• xy = k

• xy = 0

• x - y = k

Miscellaneous Exercise 4 | Q 1.12 | Page 130

Choose correct alternatives:

If h2 = ab, then slopes of lines ax2 + 2hxy + by2 = 0 are in the ratio

• 1:2

• 2:1

• 2:3

• 1:1

Miscellaneous Exercise 4 | Q 1.13 | Page 130

Choose correct alternatives:

If slope of one of the lines ax2 + 2hxy + by2 = 0 is 5 times the slope of the other, then 5h2 = ______

• ab

• 2ab

• 7ab

• 9ab

Miscellaneous Exercise 4 | Q 1.14 | Page 130

Choose correct alternatives:

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

• ± 3

• ± 5sqrt5

• 0

• ±3sqrt5

### 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]

Miscellaneous Exercise 4 | Q 1.01 | Page 130

Find the joint equation of the line:

x - y = 0 and x + y = 0

Miscellaneous Exercise 4 | Q 1.02 | Page 130

Find the joint equation of the line:

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

Miscellaneous Exercise 4 | Q 1.03 | Page 130

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

Miscellaneous Exercise 4 | Q 1.04 | Page 130

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

Miscellaneous Exercise 4 | Q 1.05 | Page 130

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

Miscellaneous Exercise 4 | Q 1.06 | Page 130

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

Miscellaneous Exercise 4 | Q 1.07 | Page 131

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

Miscellaneous Exercise 4 | Q 1.08 | Page 131

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

Miscellaneous Exercise 4 | Q 1.09 | Page 131

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

Miscellaneous Exercise 4 | Q 1.1 | Page 131

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.

Miscellaneous Exercise 4 | Q 1.11 | Page 131

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

Miscellaneous Exercise 4 | Q 2.1 | Page 131

Show that the following equations represents a pair of line:

x2 + 2xy - y2 = 0

Miscellaneous Exercise 4 | Q 2.2 | Page 131

Show that the following equations represents a pair of line:

4x2 + 4xy + y2 = 0

Miscellaneous Exercise 4 | Q 2.3 | Page 131

Show that the following equations represent a pair of line:

x2 - y2 = 0

Miscellaneous Exercise 4 | Q 2.4 | Page 131

Show that the following equations represent a pair of line:

x2 + 7xy - 2y2 = 0

Miscellaneous Exercise 4 | Q 2.5 | Page 131

Show that the following equations represent a pair of line:

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

Miscellaneous Exercise 4 | Q 3.1 | Page 131

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

6x2 - 5xy - 6y2 = 0

Miscellaneous Exercise 4 | Q 3.2 | Page 131

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

x2 - 4y2 = 0

Miscellaneous Exercise 4 | Q 3.3 | Page 131

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

3x2 - y2 = 0

Miscellaneous Exercise 4 | Q 3.4 | Page 131

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

2x2 + 2xy - y2 = 0

Miscellaneous Exercise 4 | Q 4.1 | Page 131

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

x2 + 4xy - 5y2 = 0

Miscellaneous Exercise 4 | Q 4.2 | Page 131

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

2x2 - 3xy - 9y2 = 0

Miscellaneous Exercise 4 | Q 4.3 | Page 131

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

x2 + xy - y2 = 0

Miscellaneous Exercise 4 | Q 5.1 | Page 131

Find k, if the sum of the slopes of the lines given by 3x2 + kxy - y2 = 0 is zero.

Miscellaneous Exercise 4 | Q 5.2 | Page 131

Find k, if the sum of the slopes of the lines given by x2 + kxy - 3y2 = 0 is equal to their product.

Miscellaneous Exercise 4 | Q 5.3 | Page 131

Find k, if the slope of one of the lines given by 3x2 - 4xy + ky2 = 0 is 1.

Miscellaneous Exercise 4 | Q 5.4 | Page 131

Find k, if one of the lines given by 3x2 - kxy + 5y2 = 0 is perpendicular to the line 5x + 3y = 0.

Miscellaneous Exercise 4 | Q 5.5 | Page 131

Find k if the slope of one of the lines given by 3x2 + 4xy + ky2 = 0 is three times the other.

Miscellaneous Exercise 4 | Q 5.6 | Page 131

Find k, if the slopes of lines given by kx2 + 5xy + y2 = 0 differ by 1.

Miscellaneous Exercise 4 | Q 5.7 | Page 131

Find k, if one of the lines given by 6x2 + kxy + y2 = 0 is 2x + y = 0.

Miscellaneous Exercise 4 | Q 6 | Page 131

Find the joint equation of the pair of lines which bisect angles between the lines given by x2 + 3xy + 2y2 = 0

Miscellaneous Exercise 4 | Q 7 | Page 131

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

Miscellaneous Exercise 4 | Q 8 | Page 131

Show that the lines x2 - 4xy + y2 = 0 and x + y = 10 contain the sides of an equilateral triangle. Find the area of the triangle.

Miscellaneous Exercise 4 | Q 9 | Page 131

If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is three times the other, prove that 3h2 = 4ab.

Miscellaneous Exercise 4 | Q 10 | Page 132

Find the combined equation of bisectors of angles between the lines represented by 5x2 + 6xy - y2 = 0.

Miscellaneous Exercise 4 | Q 11 | Page 132

Find an if the sum of the slope of lines represented by ax2 + 8xy + 5y2 = 0 is twice their product.

Miscellaneous Exercise 4 | Q 12 | Page 132

If the line 4x - 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0, then show that 25a + 40h + 16b = 0

Miscellaneous Exercise 4 | Q 13.1 | Page 132

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

9x2 - 6xy + y2 + 18x - 6y + 8 = 0

Miscellaneous Exercise 4 | Q 13.2 | Page 132

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

2x2 + xy - y2 + x + 4y - 3 = 0

Miscellaneous Exercise 4 | Q 13.3 | Page 132

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

Miscellaneous Exercise 4 | Q 14 | Page 132

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

Miscellaneous Exercise 4 | Q 15 | Page 132

If the lines represented by ax2 + 2hxy + by2 = 0 make angles of equal measure with the coordinate axes, then show that a ± b.

OR

Show that, one of the lines represented by ax2 + 2hxy + by2 = 0 will make an angle of the same measure with the X-axis as the other makes with the Y-axis, if a = ± b.

Miscellaneous Exercise 4 | Q 16 | Page 132

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 x2 + 2(sec 2α)xy + y2 = 0

Miscellaneous Exercise 4 | Q 17 | Page 132

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.

Miscellaneous Exercise 4 | Q 18 | Page 132

Show that the lines x2 - 4xy + y2 = 0 and the line x + y = sqrt6 form an equilateral triangle. Find its area and perimeter.

Miscellaneous Exercise 4 | Q 19 | Page 132

If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is square of the slope of the other line, show that a2b + ab2 + 8h3 = 6abh.

Miscellaneous Exercise 4 | Q 20 | Page 132

Prove that the product of length of perpendiculars drawn from P(x1, y1) to the lines represented by ax2 + 2hxy + by2 = 0 is |("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a - b")^2 + "4h"^2)|

Miscellaneous Exercise 4 | Q 21 | Page 132

Show that the difference between the slopes of the lines given by (tan2θ + cos2θ)x2 - 2xy tan θ + (sin2θ)y2 = 0 is two.

Miscellaneous Exercise 4 | Q 22 | Page 132

Find the condition that the equation ay2 + bxy + ex + dy = 0 may represent a pair of lines.

Miscellaneous Exercise 4 | Q 23 | Page 132

If the lines given by ax2 + 2hxy + by2 = 0 form an equilateral triangle with the line lx + my = 1, show that (3a + b)(a + 3b) = 4h2.

Miscellaneous Exercise 4 | Q 24 | Page 132

If the line x + 2 = 0 coincides with one of the lines represented by the equation x2 + 2xy + 4y + k = 0, then prove that k = - 4.

Miscellaneous Exercise 4 | Q 25 | Page 132

Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax2 + 2hxy + by2 = 0 is bx2 - 2hxy + ay2 = 0.

Miscellaneous Exercise 4 | Q 26 | Page 132

If equation ax2 - y2 + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.

## Chapter 4: Pair of Straight Lines

Exercise 4.1Exercise 4.2Exercise 4.3Miscellaneous Exercise 4

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

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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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