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

Answer:

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

Answer:

i) The line passing through the following points:

ii) The line passing through the following points:

iii) The line passing through the following points:

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

;

;

;

;

Answer:

i) be the slope of line joining

be the slope of line joining

, the two lines are parallel to each other.

ii) be the slope of line joining

be the slope of line joining

, the two lines are parallel to each other.

iii) be the slope of line joining

be the slope of line joining

, the two lines are perpendicular to each other.

iv) be the slope of line joining

be the slope of line joining

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:

Answer:

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.

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 is parallel to the line through ?

Answer:

be the slope of line joining

be the slope of line joining

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 is parallel to the line joining .

Answer:

be the slope of line joining

be the slope of line joining

the two lines are parallel to each other .

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

Answer:

be the slope of line joining

be the slope of line joining

the two lines are perpendicular to each other.

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

Answer:

are the vertices of a right angled triangle.

Therefore

Hence is a right angled triangle.

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

Answer:

Therefore is a rectangle.

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

Answer:

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:

be the slopes of the given lines

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 are the vertices of a parallelogram.

Answer:

are the vertices of a quadrilateral.

Therefore is a parallelogram.

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

Answer:

Slope of line joining

Slope of x-axis

If is the angle between the line and the , then

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

Answer:

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

Answer:

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

Answer:

Slope of line joining

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 are the vertices of a parallelogram.

Answer:

are the vertices of a quadrilateral.

Therefore is a parallelogram.

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

Answer:

Given points:

is a parallelogram.