Question 1: Find the slopes of the lines which make the following angles with the positive direction of x-axis:

i) ii) iii) iv)

Answer:

i) Angle with the positive direction of x-axis

ii) Angle with the positive direction of x-axis

iii) Angle with the positive direction of x-axis

iv) Angle with the positive direction of x-axis

Question 2: Find the slope of a line passing through the following points:

i) and ii) and iii) and

Answer:

i) The line passing through the following points: and

Slope

ii) The line passing through the following points: and

Slope

iii) The line passing through the following points: and

Slope

Question 3: State whether the two lines in each of the following are parallel, perpendicular or neither:

(i) Through and ; through and

(ii) Through and ; through and

(iii) Through and ; through and

(iv) Through and ); through and

Answer:

i) Let be the slope of line joining and

Slope

Let be the slope of line joining and

Slope

Since , the two lines are parallel to each other.

ii) Let be the slope of line joining and

Slope

Let be the slope of line joining and

Slope

Since , the two lines are parallel to each other.

iii) Let be the slope of line joining and

Slope

Let be the slope of line joining and

Slope

Since , the two lines are perpendicular to each other.

iv) Let be the slope of line joining and

Slope

Let be the slope of line joining and

Slope

Since and neither the two lines are neither parallel or perpendicular to each other.

Question 4: Find the slope of a line (i) which bisects the first quadrant angle (ii) which makes an angle of with the positive direction of y-axis measured anticlockwise.

Answer:

i) We know that the angle between the coordinate axes is .

The line bisects the first quadrant.

Therefore the inclination of the line with positive x-axis is

Hence the slope of line

ii) Given the line makes with positive y-axis.

Therefore the angle with positive x-axis is

Hence the slope of line

Question 5: Using the method of slope, show that the following points are collinear:

(i) (ii)

Answer:

(i) Given

Slope of

Slope of

Slope of

Since all the three lines have the same slope, they are parallel to each other. And since they have a common point, they are collinear.

(ii) Given

Slope of

Slope of

Slope of

Since all the three lines have the same slope, they are parallel to each other. And since they have a common point, they are collinear.

Question 6: What is the value of so that the line through and is parallel to the line through and ?

Answer:

Let be the slope of line joining and

Slope

Let be the slope of line joining and

Slope

Since the two lines are parallel to each other .

Question 7: What can be said regarding a line if its slope is (i) zero (ii) positive (iii) negative?

Answer:

i) If the slope

When the slope of a line is , then the line is parallel to x-axis.

ii) If the slope is positive, then is positive is acute.

Thus the line makes an acute angle with positive x-axis.

iii) If the slope is negative, then is negative is obtuse.

Thus the line makes an obtuse angle with positive x-axis.

Question 8: Show that the line joining and is parallel to the line joining and .

Answer:

Let be the slope of line joining and

Slope

Let be the slope of line joining and

Slope

Since the two lines are parallel to each other .

Question 9: Show that the line joining and is perpendicular to the line joining and .

Answer:

Let be the slope of line joining and

Let be the slope of line joining and

Since the two lines are perpendicular to each other.

Question 10: Without using Pythagoras theorem, show that the points and are the vertices of a right angled triangle.

Answer:

Given and are the vertices of a right angled triangle.

Slope of

Slope of

Slope of

Slope of Slope

Therefore

Hence is a right angled triangle.

Question 11: Prove that the points and are the vertices of a rectangle.

Answer:

Slope of

Slope of

Slope of

Slope of

Now

Similarly,

Also

Similarly,

Therefore is a rectangle.

Question 12: If the point s and lie on a line, show that:

Answer:

Given and lie on a line i.e. they are collinear.

Question 13: The slope of a line is double of the slope of another line. If tangents of the angle between them is , find the slopes of the other line.

Answer:

Let and be the slopes of the given lines

Given

Let be the angle between the lines between the two lines

Case 1: Positive sign

Case 2: Negative sign

Question 14: Consider the following population and year graph: Find the slope of the line and using it, find what will be the population in the year 2010.

Answer:

From the graph:

Slope of line

Now slope of Slope of

Cr.

Question 15: Without using the distance formula, show that points and are the vertices of a parallelogram.

Answer:

Given points and are the vertices of a quadrilateral.

Slope of

Slope of

Slope of

Slope of

Now

Similarly,

Therefore is a parallelogram.

Question 16: Find the angle between the and the line joining the points and .

Answer:

Slope of line joining and

Slope of x-axis

If is the angle between the line and the , then

Question 17: Line through the points and is perpendicular to the line through the points and . Find the value of .

Answer:

Given points

Slope of

Slope of

Given

Question 18: Find the value of for which the points and are collinear.

Answer:

Given points

Slope of

Slope of

Given

Question 19: Find the angle between x-axis and the line joining the points and .

Answer:

Slope of line joining and

Slope of x-axis

If is the angle between the line and the , then

Question 20: By using the concept of slope, show that the points and are the vertices of a parallelogram.

Answer:

Given points and are the vertices of a quadrilateral.

Slope of

Slope of

Slope of

Slope of

Now

Similarly,

Therefore is a parallelogram.

Question 21: A quadrilateral has vertices and . Show that the mid-points of the sides of this quadrilateral form a parallelogram.

Answer:

Given points: and

Let be the mid point of

Let be the mid point of

Let be the mid point of

Let be the mid point of

Slope of

Slope of

Slope of

Slope of

Since

Also since

is a parallelogram.