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

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

Find the combined equation of the following pair of lines:

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:

3y² + 7xy = 0 

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

5x² – 9y² = 0

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

x² - 4xy = 0 

Find the separate equations of the lines represented by the equation 3x² – 10xy – 8y² = 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:

x² + 2(cosec α)xy + y² = 0

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

x² + 2xy tan α - y² = 0

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 

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 

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 

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 

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

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

Find k, the slope of one of the lines given by kx² + 4xy – y² = 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 ax² + 2hxy + by² = 0 

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

If one of the lines given by ax² + 2hxy + by² = 0 is perpendicular to px + qy = 0, show that ap² + 2hpq + bq² = 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 ax² + 2hxy + by² = 0 is four times the other, show that 16h² = 25ab.

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 .

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

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

Find the value of k if lines represented by kx² + 4xy – 4y² = 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:

4x² + 5xy + y² = 0

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

2x² + 7xy + 3y² = 0

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

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

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

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.

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.

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

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

Show that the equation 2x² + xy - y² + 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:

3x² + 10xy + 3y² + 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:

x² + 3xy + 2y² + x - y + k = 0

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

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

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.

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.

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

Choose correct alternatives:

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

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

Choose correct alternatives:

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

Choose correct alternatives:

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

Choose correct alternatives:

If two lines ax² + 2hxy + by² = 0 make angles α and β with X-axis, then tan (α + β) = _____.

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 = ______.

Choose correct alternatives:

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

Choose correct alternatives:

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

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 ______.

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

Choose correct alternatives:

The combined equation of the coordinate axes is

Choose correct alternatives:

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

Choose correct alternatives:

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

Choose correct alternatives:

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

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 the lines  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:

x² + 2xy - y² = 0

Show that the following equations represents a pair of line:

4x² + 4xy + y² = 0

Show that the following equations represent a pair of line:

x² - y² = 0

Show that the following equations represent a pair of line:

x² + 7xy - 2y² = 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:

6x² - 5xy - 6y² = 0

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

x² - 4y² = 0

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

3x² - y² = 0

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

2x² + 2xy - y² = 0

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

x² + 4xy - 5y² = 0

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

2x² - 3xy - 9y² = 0

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

x² + xy - y² = 0

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

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

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

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

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

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

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

Find the joint equation of the pair of lines which bisect angles between the lines given by x² + 3xy + 2y² = 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 x² − 4xy + y² = 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 ax² + 2hxy + by² = 0 is three times the other, prove that 3h² = 4ab.

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

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

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

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

9x² - 6xy + y² + 18x - 6y + 8 = 0

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

2x² + xy - y² + 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 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.

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

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 x² - 4xy + y² = 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 ax² + 2hxy + by² = 0 is square of the slope of the other line, show that a²b + ab² + 8h³ = 6abh.

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)|`

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

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

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.

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. 

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.

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