Question 1: Find the slope and of the line:

i)

ii)

iii)

Answer:

i) Given equation is

and

ii) Given equation is

and

iii) Given equation is

and

Question 2: The equation of a line is . Find its slope and . Also, find its inclination.

Answer:

Given equation is

and

Question 3:

i) Is the line perpendicular to the line ?

ii) Is the line perpendicular to the line ?

iii) Is the line parallel to the line ?

iv) Determine so that the slope of the line through .

Answer:

i) Given equation is

Given equation is

Since , the two lines are perpendicular to each other.

ii) Given equation is

Given equation is

Since , the two lines are NOT perpendicular to each other.

iii) Given equation is

Given equation is

Since , the two lines are NOT perpendicular to each other.

Slope :

Question 4: Find the slope of the line which is parallel to:

i)

ii)

Answer:

i) Given equation is

Therefore the slope of line parallel to the given line is

ii) Given equation is

Therefore the slope of line parallel to the given line is

Question 5: Find the slope of the line which is perpendicular to:

i)

ii)

Answer:

i) Given equation is

Let the slope of line perpendicular to the given line is:

Therefore

ii) Given equation is

Let the slope of line perpendicular to the given line is:

Therefore

Question 6:

i) Lines are parallel to each other. Find the relation connecting .

ii) Lines are perpendicular to each other. Find the relation connecting .

Answer:

i) Given equation is

Given equation is

Since they are parallel,

ii) Given equation is

Slope

Given equation is

Since they are perpendicular,

Question 7: Find the value of if the lines, whose equations are are perpendicular to each other.

Answer:

Given equation is

Given equation is

Since they are perpendicular,

Question 8: The equation of a line is .

i) Find the slope of the line .

ii) Calculate the angle that the line makes with the positive direction of the .

Answer:

Given equation is

Question 9: The lines represented by are parallel. Find the value of .

Answer:

Given equation is

Given equation is

Since they are parallel,

Question 10: If the lines are perpendicular to each other, find the value of .

Given equation is

Given equation is

Since they are perpendicular,

Question 11: The line through is perpendicular to the line . Find the value of .

Answer:

Slope of

Given equation is

Since they are perpendicular,

Question 12: Find the equation of the line passing through and parallel to:

i)

ii)

Answer:

i) Line parallel to has a slope of

Equation of a line with slope and passing through is

ii) Line parallel to has a slope of

Equation of a line with slope and passing through is

Question 13:

i) Find the equation of the line passing through and parallel to .

ii) Find the equation of the line parallel to the line and passing through the point .

Answer:

i) Given Point

Given equation is

Equation of a line with slope and passing through is

ii) Given Point

Given equation is

Equation of a line with slope and passing through is

Question 14: Find the equation of the line passing through and perpendicular to .

Answer:

Given Point

Given equation is

Therefore slope of the new line

Equation of a line with slope and passing through is

Question 15: Find the equation of the perpendicular bisector of the line segment obtained on joining the points .

Answer:

Let P be the bisector of the points

Let the coordinates of

Therefore

Slope of the line joining the two given points

Therefore the slope of the line perpendicular to the line joining

Equation of a line with slope and passing through is

Question 16: are the vertices of rhombus . Find the equations of the diagonals .

Answer:

Slope of

Slope of

Equation of a line with slope and passing through is

Midpoint of BD

Equation of AC

Question 17: are the vertices of a square . Find the equations of the diagonals .

Answer:

Slope of

Slope of

Equation of AC :

Midpoint of BD

Equation of BD

Question 18: are the vertices of triangle . Find the equation of:

i) The median of the triangle through .

ii) The altitude of the triangle through .

iii) The line through and parallel to .

Answer:

i) Midpoint of

Slope of

Equation of BD

ii) Slope of

Slope of line perpendicular to this

Equation of line

iii) Slope of

Slope of line parallel to this

Equation of line

Question 19:

i) Write down the equation of the line , through and perpendicular to the line .

ii) meets the and the at . write down the co-ordinates of . Calculate the area of triangle , where is origin.

Answer:

Given equation is

Therefore slope of the new line

Equation of a line with slope and passing through is

ii) Equation of is

When . Therefore

When . Therefore

Area of the triangle sq. units.

Question 20: The line meets . write the co-ordinates of . Determine the equation of the line through and perpendicular to .

Answer:

Given equation

When

Therefore

Given equation is

Therefore slope of the line perpendicular to this line

Equation of a line with slope and passing through is

Question 21: The point is the foot of perpendicular from to the line whose equation Determine:

i) The equation of the line

ii) The co-ordinates of

Answer:

i) Given equation is

Therefore slope of the line perpendicular to this line

Equation of AP with slope and passing through is

ii) The coordinate of P is the intersection of the two lines:

and .

Solving the two equations, we get . Therefor

Question 22: The points are respectively. Find the equations of . If cuts the at cuts the at , find the co-ordinates of .

Answer:

Given points

Slope of

Slope of

Equation of AB with slope and passing through is

Equation of AB with slope and passing through is

Intercept of on

Intercept of on

Question 23: Find the value of a for the points are collinear. Hence, find the equation of the line.

Answer:

Given points

Slope of

Slope of

Because are collinear: