Question 1: Find the angles between each of the following pairs of straight lines:

Answer:

be the slope of the straight lines respectively.

be the angle between the lines. Then,

be the slope of the straight lines respectively.

be the angle between the lines. Then,

be the slope of the straight lines respectively.

Since

Therefore the given lines are perpendicular.

be the slope of the straight lines respectively.

be the angle between the lines. Then,

be the angle between the lines. Then,

Question 2: Find the acute angle between the lines .

Answer:

be the slope of the straight lines respectively.

be the angle between the lines. Then,

Question 3: Prove that the points are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.

Answer:

Let the vertices are

Therefore

Hence is a parallelogram.

be the slope of the straight lines respectively.

is to y-axis

Question 4: Find the angle between the line joining the points and the line .

Answer:

Let the points be

be the angle between the lines. Then,

Question 5: If is the angle which the straight line joining the points subtends at the origin, prove that

Answer:

Let the two points be

We know

Question 6: Prove that the straight lines form an isosceles triangle whose vertical angle is

Answer:

Given Lines:

Line i) :

Line ii) :

Line iii) :

is the angle between line i) and line ii)

is the angle between line ii) and line iii)

is the angle between line iii) and line i)

Therefore the triangle is an isosceles as

The vertical angle is

Question 7: Find the angle between the lines .

Answer:

Given: Line i) : . This line is to y-axis.

Line ii) . This line is to x-axis.

Question 8: Find the tangent of the angle between the lines which have intercepts on the axes respectively.

Answer:

Equation of the line with intercepts :

… … … … … i)

Equation of the line with intercepts :

… … … … … ii)

Slope of line i)

Slope of line ii)

Question 9: Show that the line is perpendicular to the line for all non-zero real values of .

Answer:

Line i) :

Line ii) :

Now,

Therefore the two lines are perpendicular to each other.

Question 10: Show that the tangent of an angle between the lines is

Answer:

Given lines:

… … … … … i)

… … … … … ii)

Slope of line i)

Slope of line ii)