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

(i) and

(ii) and

(iii) and

(iv) and

v) and

Answer:

(i) Let and be the slope of the straight lines and respectively.

Then and

Let be the angle between the lines. Then,

Thus, the acute angle between the lines is of

(ii) Let and be the slope of the straight lines and respectively.

Then and

Let be the angle between the lines. Then,

Thus, the acute angle between the lines is of

(iii) Let and be the slope of the straight lines and respectively.

Then and

Since

Therefore the given lines are perpendicular.

Thus, the acute angle between the lines is of

(iv) Let and be the slope of the straight lines and respectively.

Then and

Let be the angle between the lines. Then,

Thus, the acute angle between the lines is of

v) and

Then and

Let be the angle between the lines. Then,

Thus, the acute angle between the lines is of

Question 2: Find the acute angle between the lines and .

Answer:

Let and be the slope of the straight lines and respectively.

Then and

Let be the angle between the lines. Then,

Thus, the acute angle between the lines is of

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

Answer:

Let the vertices are and

Slope of

Slope of

Slope of

Slope of

Therefore and

Hence is a parallelogram.

Let and be the slope of the straight lines and respectively.

Then is to y-axis

Thus, the acute angle between the diagonals is of

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

Answer:

Let the points be

Slope of

Slope of

Let be the angle between the lines. Then,

Thus, the acute angle between the lines is of

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

Answer:

Let the two points be and

Slope of

Slope of

We know

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

Answer:

Given Lines:

Line i) :

Line ii) :

Line iii) :

Let is the angle between line i) and line ii)

Let is the angle between line ii) and line iii)

Let 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 and .

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 and 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 and is

Answer:

Given lines:

… … … … … i)

… … … … … ii)

Slope of line i)

Slope of line ii)